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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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
Description
The largest part of an integer partition.
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 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3 = 4 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4 = 5 - 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4 = 5 - 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]
=> 5 = 6 - 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]
=> 6 = 7 - 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3 = 4 - 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4 = 5 - 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 3 = 4 - 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 4 = 5 - 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4 = 5 - 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]
=> 5 = 6 - 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]
=> 6 = 7 - 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 4 = 5 - 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 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]
=> 5 = 6 - 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4 = 5 - 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]
=> 5 = 6 - 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]
=> 6 = 7 - 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4 = 5 - 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]
=> 5 = 6 - 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the 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: 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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[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
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 5
[[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
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[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
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[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
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[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
[[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
Description
The length of the partition.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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
[[3,3]]
=> ([(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
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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],[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
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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
[[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
[[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
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[[1,1,1,2]]
=> ([(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
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [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
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
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].
Matching statistic: St000384
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: 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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[1,2,3],[2],[5]]
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> ? = 6
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: 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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
[[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
[[1,2,3],[2],[5]]
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> ? = 6
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: St000380
(load all 2 compositions to match this statistic)
(load all 2 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 = 2 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1],[2]]
=> ([],1)
=> [1]
=> 2 = 1 + 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 4 + 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 5 + 1
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 2 = 1 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 4 + 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 5 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 4 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6 = 5 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 5 + 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 2 = 1 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 5 + 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 2 = 1 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 2 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 3 + 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 2 = 1 + 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 5 + 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]
=> 7 = 6 + 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 4 + 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 5 + 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]
=> 7 = 6 + 1
[[1,2,3],[2],[5]]
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> ? = 6 + 1
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: 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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[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
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5
[[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
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[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
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[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
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[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
[[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
[[2,2,2,2,2,2,2]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 8
[[1,8]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 8
[[1,2,3],[2],[5]]
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> ([(2,11),(2,12),(3,4),(3,12),(4,11),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,12),(8,9),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 6
[[1,2],[2,5],[4]]
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(3,10),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 6
[[1,2],[3,3],[5]]
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(3,10),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 6
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,4],[3,3],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 7
[[1,2,3],[3,3],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 7
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[2,2,2,2,2,2],[3]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 8
[[1,2,2,2,2,2,2,2]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 8
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 7
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
=> ? = 6
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
=> ? = 6
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 7
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: St001039
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 96%●distinct values known / distinct values provided: 88%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 96%●distinct values known / distinct values provided: 88%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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
[[3,3]]
=> ([(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
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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],[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
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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
[[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
[[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
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,2]]
=> ([(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
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [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
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[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
[[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
[[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
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 6
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[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
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1],[2],[3],[4]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1],[2],[3],[4]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1],[2,2],[3,3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1],[2],[3],[4],[5]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2],[3],[4]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1],[2,2],[3],[4]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2,2,2],[3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1],[2,2],[3,3]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2,2,2,2]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1,1,1,1,1]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[[1],[2],[3],[4],[5],[6]]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
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
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[1,1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[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
[[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
[[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
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[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
[[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
[[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
[[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
[[1,1],[3]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[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
[[1],[2],[3]]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[[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
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[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
[[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
[[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
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[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
[[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
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,3,4],[3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[2,2,2,2,3]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[2,2,2,3],[3]]
=> ([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[2],[7]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,3],[3,5]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,3,4],[3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[2,2,2,2],[4]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,4],[2,3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,2],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,2,2,2,2,3]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,2,2,3],[3]]
=> ([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,2],[7]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,1,2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,6],[2]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,3],[2],[6]]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,3],[3,5]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,3],[2],[5]]
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 6
[[2,2,2],[3],[5]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,1],[3,5],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,2],[2,5],[4]]
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 6
[[1,2],[3,3],[5]]
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 6
[[1,1,1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,3,4],[3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,2,2,2],[4]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,1,2,4],[2,3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,3,4],[2,3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,2,3],[2],[4]]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,2],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,4],[3,3],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,2,4],[2,3],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,2,3],[3,3],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,1,2,2,2,2,3]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,1,2,2,2,3],[3]]
=> ([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,2,2,2,2,3],[2]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[[1,2,2,2,3],[2,3]]
=> ([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,1,2,3],[3,3,3]]
=> ([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,1,1],[2,2,2,3],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
Description
The last entry in the first row of a standard tableau.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001622The number of join-irreducible elements of a lattice. 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. St000474Dyson's crank of a partition. 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. St001286The annihilation number 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. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. 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. St001644The dimension of a graph. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St000822The Hadwiger number of the graph. St001316The domatic number 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. St001875The number of simple modules with projective dimension at most 1. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000907The number of maximal antichains of minimal length in a poset. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. 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. St000327The number of cover relations in a poset. St001621The number of atoms of a lattice. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. 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.
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