searching the database
Your data matches 116 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000112
St000112: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[2,2]]
=> 2
[[1],[2]]
=> 0
[[1,3]]
=> 2
[[2,3]]
=> 3
[[3,3]]
=> 4
[[1],[3]]
=> 1
[[2],[3]]
=> 2
[[1,1,2]]
=> 1
[[1,2,2]]
=> 2
[[2,2,2]]
=> 3
[[1,1],[2]]
=> 0
[[1,2],[2]]
=> 1
[[1,4]]
=> 3
[[2,4]]
=> 4
[[3,4]]
=> 5
[[4,4]]
=> 6
[[1],[4]]
=> 2
[[2],[4]]
=> 3
[[3],[4]]
=> 4
[[1,1,3]]
=> 2
[[1,2,3]]
=> 3
[[1,3,3]]
=> 4
[[2,2,3]]
=> 4
[[2,3,3]]
=> 5
[[3,3,3]]
=> 6
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 2
[[1,3],[2]]
=> 2
[[1,3],[3]]
=> 3
[[2,2],[3]]
=> 3
[[2,3],[3]]
=> 4
[[1],[2],[3]]
=> 0
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 2
[[1,2,2,2]]
=> 3
[[2,2,2,2]]
=> 4
[[1,1,1],[2]]
=> 0
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 2
[[1,1],[2,2]]
=> 0
[[1,5]]
=> 4
[[2,5]]
=> 5
[[3,5]]
=> 6
[[4,5]]
=> 7
[[5,5]]
=> 8
[[1],[5]]
=> 3
[[2],[5]]
=> 4
[[3],[5]]
=> 5
[[4],[5]]
=> 6
Description
The sum of the entries reduced by the index of their row in a semistandard tableau.
This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.
Matching statistic: St000093
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> 7 = 6 + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> 8 = 7 + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> 9 = 8 + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 7 = 6 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000380
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1],[2]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 3 + 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 4 + 2
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[1,1],[2]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 4 + 2
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 7 = 5 + 2
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 8 = 6 + 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 3 + 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 4 + 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 3 + 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 4 + 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 4 + 2
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 7 = 5 + 2
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 8 = 6 + 2
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 4 + 2
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 4 + 2
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 4 + 2
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 7 = 5 + 2
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 8 = 6 + 2
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> 9 = 7 + 2
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> 10 = 8 + 2
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 4 + 2
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 7 = 5 + 2
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 8 = 6 + 2
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St001392
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1],[2]]
=> ([],1)
=> [1]
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[1,1],[2]]
=> ([],1)
=> [1]
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 5
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 6
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 5
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 6
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 4
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 0
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 0
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 5
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 6
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> 7
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> 8
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 5
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 6
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,8,9}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,8,9}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {7,8,9}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {8,9,10}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {8,9,10}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {8,9,10}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12}
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[1,3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,3,4],[3]]
=> ([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,3,4],[4]]
=> ([(0,2),(0,3),(1,8),(1,10),(2,1),(3,5),(3,7),(4,26),(5,22),(6,20),(7,12),(7,22),(8,21),(9,18),(9,19),(10,21),(10,25),(11,14),(11,15),(12,9),(12,24),(12,25),(13,27),(14,27),(15,27),(16,13),(17,14),(18,16),(19,17),(20,11),(20,17),(21,4),(21,23),(22,6),(22,24),(23,16),(23,26),(24,19),(24,20),(25,18),(25,23),(26,13),(26,15)],28)
=> [9,7,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,4,4],[3]]
=> ([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,4,4],[4]]
=> ([(0,11),(0,13),(1,19),(2,20),(3,26),(4,17),(4,22),(5,16),(6,14),(6,21),(7,14),(7,15),(8,9),(9,1),(9,15),(10,4),(10,25),(11,8),(12,6),(12,7),(12,16),(13,5),(13,12),(14,23),(15,19),(15,23),(16,10),(16,21),(17,26),(17,27),(18,20),(19,22),(19,24),(21,25),(22,27),(23,24),(24,27),(25,3),(25,17),(26,2),(26,18),(27,18)],28)
=> [10,8,5,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,3,4],[4]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,4,4],[4]]
=> ([(0,1),(0,2),(1,10),(1,12),(2,9),(2,11),(3,35),(4,33),(5,42),(6,32),(7,31),(8,15),(8,34),(9,36),(10,37),(11,16),(11,36),(12,17),(12,18),(12,37),(13,21),(13,23),(14,24),(14,38),(15,22),(15,28),(16,14),(16,40),(16,41),(17,30),(17,41),(18,26),(18,30),(20,44),(21,44),(22,43),(23,4),(23,44),(24,25),(25,23),(26,34),(27,20),(27,43),(28,35),(28,43),(29,31),(30,5),(30,39),(31,19),(32,13),(32,25),(33,19),(34,3),(34,28),(35,7),(35,29),(36,6),(36,40),(37,8),(37,26),(38,22),(38,27),(39,27),(39,42),(40,24),(40,32),(41,38),(41,39),(42,20),(42,21),(43,29),(44,33)],45)
=> [11,9,7,7,5,3,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,3],[4,4]]
=> ([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[3,3],[4,4]]
=> ([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> [9,5,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12}
[[2,2,3,3,3]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {7,8,8,9,10}
[[2,3,3,3,3]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {7,8,8,9,10}
[[3,3,3,3,3]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {7,8,8,9,10}
[[2,2,3,3],[3]]
=> ([(0,10),(0,11),(1,12),(2,17),(3,13),(4,14),(5,9),(5,12),(6,5),(7,3),(8,1),(8,17),(9,4),(9,15),(10,6),(11,2),(11,8),(12,15),(13,16),(14,16),(15,13),(15,14),(17,7)],18)
=> [8,6,4]
=> ? ∊ {7,8,8,9,10}
[[2,3,3,3],[3]]
=> ([(0,13),(0,14),(1,16),(2,15),(3,17),(4,19),(5,18),(6,12),(6,16),(7,6),(8,2),(8,22),(9,1),(9,22),(10,4),(11,3),(11,23),(12,5),(12,20),(13,7),(14,8),(14,9),(15,23),(16,20),(17,21),(18,21),(20,17),(20,18),(21,19),(22,11),(22,15),(23,10)],24)
=> [9,7,5,3]
=> ? ∊ {7,8,8,9,10}
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000147
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 7 = 6 + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> 8 = 7 + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> 9 = 8 + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {7,8,9} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {8,9,10} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4],[3]]
=> ([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4],[4]]
=> ([(0,2),(0,3),(1,8),(1,10),(2,1),(3,5),(3,7),(4,26),(5,22),(6,20),(7,12),(7,22),(8,21),(9,18),(9,19),(10,21),(10,25),(11,14),(11,15),(12,9),(12,24),(12,25),(13,27),(14,27),(15,27),(16,13),(17,14),(18,16),(19,17),(20,11),(20,17),(21,4),(21,23),(22,6),(22,24),(23,16),(23,26),(24,19),(24,20),(25,18),(25,23),(26,13),(26,15)],28)
=> [9,7,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4],[3]]
=> ([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4],[4]]
=> ([(0,11),(0,13),(1,19),(2,20),(3,26),(4,17),(4,22),(5,16),(6,14),(6,21),(7,14),(7,15),(8,9),(9,1),(9,15),(10,4),(10,25),(11,8),(12,6),(12,7),(12,16),(13,5),(13,12),(14,23),(15,19),(15,23),(16,10),(16,21),(17,26),(17,27),(18,20),(19,22),(19,24),(21,25),(22,27),(23,24),(24,27),(25,3),(25,17),(26,2),(26,18),(27,18)],28)
=> [10,8,5,4,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,4],[4]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,4,4],[4]]
=> ([(0,1),(0,2),(1,10),(1,12),(2,9),(2,11),(3,35),(4,33),(5,42),(6,32),(7,31),(8,15),(8,34),(9,36),(10,37),(11,16),(11,36),(12,17),(12,18),(12,37),(13,21),(13,23),(14,24),(14,38),(15,22),(15,28),(16,14),(16,40),(16,41),(17,30),(17,41),(18,26),(18,30),(20,44),(21,44),(22,43),(23,4),(23,44),(24,25),(25,23),(26,34),(27,20),(27,43),(28,35),(28,43),(29,31),(30,5),(30,39),(31,19),(32,13),(32,25),(33,19),(34,3),(34,28),(35,7),(35,29),(36,6),(36,40),(37,8),(37,26),(38,22),(38,27),(39,27),(39,42),(40,24),(40,32),(41,38),(41,39),(42,20),(42,21),(43,29),(44,33)],45)
=> [11,9,7,7,5,3,3]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[4,4]]
=> ([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[4,4]]
=> ([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> [9,5,5,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,3,3]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {7,8,8,9,10} + 1
[[2,3,3,3,3]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {7,8,8,9,10} + 1
[[3,3,3,3,3]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {7,8,8,9,10} + 1
[[2,2,3,3],[3]]
=> ([(0,10),(0,11),(1,12),(2,17),(3,13),(4,14),(5,9),(5,12),(6,5),(7,3),(8,1),(8,17),(9,4),(9,15),(10,6),(11,2),(11,8),(12,15),(13,16),(14,16),(15,13),(15,14),(17,7)],18)
=> [8,6,4]
=> ? ∊ {7,8,8,9,10} + 1
[[2,3,3,3],[3]]
=> ([(0,13),(0,14),(1,16),(2,15),(3,17),(4,19),(5,18),(6,12),(6,16),(7,6),(8,2),(8,22),(9,1),(9,22),(10,4),(11,3),(11,23),(12,5),(12,20),(13,7),(14,8),(14,9),(15,23),(16,20),(17,21),(18,21),(20,17),(20,18),(21,19),(22,11),(22,15),(23,10)],24)
=> [9,7,5,3]
=> ? ∊ {7,8,8,9,10} + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 85%●distinct values known / distinct values provided: 69%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [2,2,2,1,1,1]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [2,2,2,1,1,1,1]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [2,2,2,1,1,1]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [2,2,2,1,1,1,1]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> [2,2,2,1,1,1]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> 7 = 6 + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> 8 = 7 + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> 9 = 8 + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [2,2,2,1,1,1]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [2,2,2,1,1,1,1]
=> 7 = 6 + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [3,3,3,3,2,2,1,1,1]
=> ? ∊ {7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [3,3,3,3,2,2,1,1,1,1]
=> ? ∊ {7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ? ∊ {7,8,9} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [3,3,3,2,2,2,1,1,1]
=> ? ∊ {8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [3,3,3,2,2,2,2,1,1,1]
=> ? ∊ {8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [3,3,3,2,2,2,2,1,1,1,1]
=> ? ∊ {8,9,10} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [4,4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [4,4,4,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> [5,4,4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> [5,4,4,4,4,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> [5,4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> [5,4,4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> [5,4,4,4,4,3,3,2,2,1,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [3,3,3,3,2,2,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [3,3,3,3,2,2,2,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [3,3,3,3,2,2,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> [5,4,4,4,3,3,2,2,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [3,3,3,3,2,2,1,1,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> [6,6,6,5,5,3,3,2,2,1,1]
=> ? ∊ {7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [3,3,3,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [3,3,3,3,2,2,1,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [4,4,4,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [4,4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> [5,4,4,4,3,3,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> [5,4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> [5,4,4,4,4,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> [5,4,4,4,4,3,3,2,2,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> [5,4,4,4,4,3,3,2,2,1,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4],[3]]
=> ([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [5,4,4,3,3,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,4],[4]]
=> ([(0,2),(0,3),(1,8),(1,10),(2,1),(3,5),(3,7),(4,26),(5,22),(6,20),(7,12),(7,22),(8,21),(9,18),(9,19),(10,21),(10,25),(11,14),(11,15),(12,9),(12,24),(12,25),(13,27),(14,27),(15,27),(16,13),(17,14),(18,16),(19,17),(20,11),(20,17),(21,4),(21,23),(22,6),(22,24),(23,16),(23,26),(24,19),(24,20),(25,18),(25,23),(26,13),(26,15)],28)
=> [9,7,6,4,2]
=> [5,5,4,4,3,3,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4],[3]]
=> ([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,4,4],[4]]
=> ([(0,11),(0,13),(1,19),(2,20),(3,26),(4,17),(4,22),(5,16),(6,14),(6,21),(7,14),(7,15),(8,9),(9,1),(9,15),(10,4),(10,25),(11,8),(12,6),(12,7),(12,16),(13,5),(13,12),(14,23),(15,19),(15,23),(16,10),(16,21),(17,26),(17,27),(18,20),(19,22),(19,24),(21,25),(22,27),(23,24),(24,27),(25,3),(25,17),(26,2),(26,18),(27,18)],28)
=> [10,8,5,4,1]
=> [5,4,4,4,3,2,2,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3,4],[4]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> [5,5,4,4,3,3,2,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,4,4],[4]]
=> ([(0,1),(0,2),(1,10),(1,12),(2,9),(2,11),(3,35),(4,33),(5,42),(6,32),(7,31),(8,15),(8,34),(9,36),(10,37),(11,16),(11,36),(12,17),(12,18),(12,37),(13,21),(13,23),(14,24),(14,38),(15,22),(15,28),(16,14),(16,40),(16,41),(17,30),(17,41),(18,26),(18,30),(20,44),(21,44),(22,43),(23,4),(23,44),(24,25),(25,23),(26,34),(27,20),(27,43),(28,35),(28,43),(29,31),(30,5),(30,39),(31,19),(32,13),(32,25),(33,19),(34,3),(34,28),(35,7),(35,29),(36,6),(36,40),(37,8),(37,26),(38,22),(38,27),(39,27),(39,42),(40,24),(40,32),(41,38),(41,39),(42,20),(42,21),(43,29),(44,33)],45)
=> [11,9,7,7,5,3,3]
=> [7,7,7,5,5,4,4,2,2,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[4,4]]
=> ([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> [4,3,3,3,3,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[4,4]]
=> ([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> [9,5,5,1]
=> [4,3,3,3,3,1,1,1,1]
=> ? ∊ {7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,3,3]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [3,3,3,2,2,2,1,1,1]
=> ? ∊ {7,8,8,9,10} + 1
[[2,3,3,3,3]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [3,3,3,2,2,2,2,1,1,1]
=> ? ∊ {7,8,8,9,10} + 1
[[3,3,3,3,3]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [3,3,3,2,2,2,2,1,1,1,1]
=> ? ∊ {7,8,8,9,10} + 1
[[2,2,3,3],[3]]
=> ([(0,10),(0,11),(1,12),(2,17),(3,13),(4,14),(5,9),(5,12),(6,5),(7,3),(8,1),(8,17),(9,4),(9,15),(10,6),(11,2),(11,8),(12,15),(13,16),(14,16),(15,13),(15,14),(17,7)],18)
=> [8,6,4]
=> [3,3,3,3,2,2,1,1]
=> ? ∊ {7,8,8,9,10} + 1
[[2,3,3,3],[3]]
=> ([(0,13),(0,14),(1,16),(2,15),(3,17),(4,19),(5,18),(6,12),(6,16),(7,6),(8,2),(8,22),(9,1),(9,22),(10,4),(11,3),(11,23),(12,5),(12,20),(13,7),(14,8),(14,9),(15,23),(16,20),(17,21),(18,21),(20,17),(20,18),(21,19),(22,11),(22,15),(23,10)],24)
=> [9,7,5,3]
=> [4,4,4,3,3,2,2,1,1]
=> ? ∊ {7,8,8,9,10} + 1
Description
The length of the partition.
Matching statistic: St000384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 73%●distinct values known / distinct values provided: 62%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 73%●distinct values known / distinct values provided: 62%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 7 = 6 + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {7,8} + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {7,8} + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {6,7,8} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {6,7,8} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> ? ∊ {6,7,8} + 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? ∊ {7,7,8,8,9,10} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {7,7,8,8,9,10} + 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {7,7,8,8,9,10} + 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[5]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000784
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 73%●distinct values known / distinct values provided: 62%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 73%●distinct values known / distinct values provided: 62%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 7 = 6 + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {7,8} + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {7,8} + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {6,7,8} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {6,7,8} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> ? ∊ {6,7,8} + 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? ∊ {7,7,8,8,9,10} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> ? ∊ {7,7,8,8,9,10} + 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {7,7,8,8,9,10} + 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {7,7,8,8,9,10} + 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[5]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> ? ∊ {6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> ? ∊ {6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000645
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 70%●distinct values known / distinct values provided: 54%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 70%●distinct values known / distinct values provided: 54%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,8} + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,8} + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,7,8} + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,8} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,7,8} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,6,7,8} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,7,8} + 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5],[5]]
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[5]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000676
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 70%●distinct values known / distinct values provided: 54%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 70%●distinct values known / distinct values provided: 54%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,8} + 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,8} + 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,8} + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {6,6,7,7,7,8,9} + 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,7,8} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,7,8} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,7,8} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,7,8} + 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,7,7,8,8,9,10} + 1
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> [10,7,6,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> [11,8,6,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> [11,8,6,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> [12,9,7,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> [13,9,7,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,5],[5]]
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,3],[5]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,4],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[4]]
=> ([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[2,5],[5]]
=> ([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,3],[5]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,4],[5]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[4]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[3,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,23),(5,24),(6,17),(7,22),(8,13),(8,24),(9,10),(9,27),(10,26),(11,16),(11,20),(12,19),(12,27),(13,18),(13,19),(15,28),(16,4),(16,28),(17,7),(18,17),(19,25),(20,22),(20,28),(21,14),(22,21),(23,14),(24,6),(24,18),(25,15),(25,20),(26,15),(26,16),(27,11),(27,25),(27,26),(28,21),(28,23)],29)
=> [10,8,6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,4],[5]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[4,5],[5]]
=> ([(0,2),(0,3),(1,9),(1,15),(2,1),(3,7),(3,8),(4,30),(5,31),(6,23),(7,16),(7,37),(8,10),(8,37),(9,11),(9,36),(10,34),(11,35),(12,25),(12,29),(13,19),(13,22),(14,21),(14,27),(15,26),(15,36),(16,26),(16,33),(17,39),(18,38),(19,12),(19,38),(20,6),(21,4),(21,39),(22,5),(22,38),(24,23),(25,28),(26,32),(27,25),(27,39),(28,24),(29,20),(30,24),(31,20),(32,17),(32,27),(33,18),(33,19),(34,18),(34,22),(35,17),(35,21),(36,14),(36,32),(36,35),(37,13),(37,33),(37,34),(38,29),(38,31),(39,28),(39,30)],40)
=> [11,9,7,5,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,10,11,12} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,11,12} + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000393The number of strictly increasing runs in a binary word. St000734The last entry in the first row of a standard tableau. St001622The number of join-irreducible elements of a lattice. St000026The position of the first return of a Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000474Dyson's crank of a partition. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001512The minimum rank of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001093The detour number of a graph. St001286The annihilation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001829The common independence number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000172The Grundy number of a graph. St000722The number of different neighbourhoods in a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001883The mutual visibility number of a graph. St001782The order of rowmotion on the set of order ideals of a poset. St000080The rank of the poset. St000189The number of elements in the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001480The number of simple summands of the module J^2/J^3. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001644The dimension of a graph. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St001828The Euler characteristic of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000741The Colin de Verdière graph invariant. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000286The number of connected components of the complement of a graph. St000822The Hadwiger number of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001963The tree-depth of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001812The biclique partition number of a graph. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!