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Your data matches 133 different statistics following compositions of up to 3 maps.
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Matching statistic: St001392
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The largest part of an integer partition.
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: St000010
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The length of the partition.
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
[[2,2,2,2,2,2,2]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 7 + 1
[[1,2,2,2,2,2,2,2]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? ∊ {7,8} + 1
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([],9)
=> ? ∊ {7,8} + 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: 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: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
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
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 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
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: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
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
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4 = 3 + 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
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: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
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
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[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
[[1,1,3,3]]
=> ([(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
[[2,2,3,3]]
=> ([(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
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: St000734
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1 = 0 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5 = 4 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[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)
=> [6,3]
=> [[1,2,3,4,5,6],[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)
=> [7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> 7 = 6 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5 = 4 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5 = 4 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[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)
=> [6,3]
=> [[1,2,3,4,5,6],[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)
=> [7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> 7 = 6 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 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]
=> [[1,2,3,4,5],[6,7,8]]
=> 5 = 4 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [[1]]
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 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]
=> [[1,2,3,4,5]]
=> 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]
=> [[1,2,3,4,5,6],[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)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? ∊ {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]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? ∊ {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]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? ∊ {6,7,8} + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[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)
=> [6,3]
=> [[1,2,3,4,5,6],[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)
=> [7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> 7 = 6 + 1
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5 = 4 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? ∊ {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]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? ∊ {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]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? ∊ {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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? ∊ {6,6,7,8} + 1
Description
The last entry in the first row of a standard tableau.
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: 89% ●values known / values provided: 95%●distinct values known / distinct values provided: 89%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 95%●distinct values known / distinct values provided: 89%
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
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[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
[[1,1,3,3]]
=> ([(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
[[2,2,3,3]]
=> ([(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
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 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 123 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. St001622The number of join-irreducible elements of a lattice. St000026The position of the first return of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. 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. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St000377The dinv defect of an integer partition. St001340The cardinality of a minimal non-edge isolating set of a graph. St001512The minimum rank of a graph. St000273The domination number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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. St000474Dyson's crank of a partition. 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. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000172The Grundy number of a graph. St000528The height of a poset. St000722The number of different neighbourhoods in a graph. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001644The dimension of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000080The rank of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St001820The size of the image of the pop stack sorting operator. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001664The number of non-isomorphic subposets of a poset. St001720The minimal length of a chain of small intervals in a lattice. St001782The order of rowmotion on the set of order ideals of a poset. St000744The length of the path to the largest entry in a standard Young tableau. 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. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001962The proper pathwidth of a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. 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. 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the 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. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. 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). 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. 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. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
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