Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => 2
[[2,2]]
 => [2]
 => 1
[[1],[2]]
 => [1,1]
 => 2
[[1,3]]
 => [1,1]
 => 2
[[2,3]]
 => [1,1]
 => 2
[[3,3]]
 => [2]
 => 1
[[1],[3]]
 => [1,1]
 => 2
[[2],[3]]
 => [1,1]
 => 2
[[1,1,2]]
 => [2,1]
 => 2
[[1,2,2]]
 => [2,1]
 => 2
[[2,2,2]]
 => [3]
 => 1
[[1,1],[2]]
 => [2,1]
 => 2
[[1,2],[2]]
 => [2,1]
 => 2
[[1,4]]
 => [1,1]
 => 2
[[2,4]]
 => [1,1]
 => 2
[[3,4]]
 => [1,1]
 => 2
[[4,4]]
 => [2]
 => 1
[[1],[4]]
 => [1,1]
 => 2
[[2],[4]]
 => [1,1]
 => 2
[[3],[4]]
 => [1,1]
 => 2
[[1,1,3]]
 => [2,1]
 => 2
[[1,2,3]]
 => [1,1,1]
 => 3
[[1,3,3]]
 => [2,1]
 => 2
[[2,2,3]]
 => [2,1]
 => 2
[[2,3,3]]
 => [2,1]
 => 2
[[3,3,3]]
 => [3]
 => 1
[[1,1],[3]]
 => [2,1]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => 3
[[1,3],[3]]
 => [2,1]
 => 2
[[2,2],[3]]
 => [2,1]
 => 2
[[2,3],[3]]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 3
[[1,1,1,2]]
 => [3,1]
 => 2
[[1,1,2,2]]
 => [2,2]
 => 2
[[1,2,2,2]]
 => [3,1]
 => 2
[[2,2,2,2]]
 => [4]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => 2
[[1,5]]
 => [1,1]
 => 2
[[2,5]]
 => [1,1]
 => 2
[[3,5]]
 => [1,1]
 => 2
[[4,5]]
 => [1,1]
 => 2
[[5,5]]
 => [2]
 => 1
[[1],[5]]
 => [1,1]
 => 2
[[2],[5]]
 => [1,1]
 => 2
[[3],[5]]
 => [1,1]
 => 2
[[4],[5]]
 => [1,1]
 => 2
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [2]
 => 2
[[2,2]]
 => [2]
 => [1,1]
 => 1
[[1],[2]]
 => [1,1]
 => [2]
 => 2
[[1,3]]
 => [1,1]
 => [2]
 => 2
[[2,3]]
 => [1,1]
 => [2]
 => 2
[[3,3]]
 => [2]
 => [1,1]
 => 1
[[1],[3]]
 => [1,1]
 => [2]
 => 2
[[2],[3]]
 => [1,1]
 => [2]
 => 2
[[1,1,2]]
 => [2,1]
 => [2,1]
 => 2
[[1,2,2]]
 => [2,1]
 => [2,1]
 => 2
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[2]]
 => [2,1]
 => [2,1]
 => 2
[[1,2],[2]]
 => [2,1]
 => [2,1]
 => 2
[[1,4]]
 => [1,1]
 => [2]
 => 2
[[2,4]]
 => [1,1]
 => [2]
 => 2
[[3,4]]
 => [1,1]
 => [2]
 => 2
[[4,4]]
 => [2]
 => [1,1]
 => 1
[[1],[4]]
 => [1,1]
 => [2]
 => 2
[[2],[4]]
 => [1,1]
 => [2]
 => 2
[[3],[4]]
 => [1,1]
 => [2]
 => 2
[[1,1,3]]
 => [2,1]
 => [2,1]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [3]
 => 3
[[1,3,3]]
 => [2,1]
 => [2,1]
 => 2
[[2,2,3]]
 => [2,1]
 => [2,1]
 => 2
[[2,3,3]]
 => [2,1]
 => [2,1]
 => 2
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[3]]
 => [2,1]
 => [2,1]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [3]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [3]
 => 3
[[1,3],[3]]
 => [2,1]
 => [2,1]
 => 2
[[2,2],[3]]
 => [2,1]
 => [2,1]
 => 2
[[2,3],[3]]
 => [2,1]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [2,2]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [2,2]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2,2]
 => 2
[[1,5]]
 => [1,1]
 => [2]
 => 2
[[2,5]]
 => [1,1]
 => [2]
 => 2
[[3,5]]
 => [1,1]
 => [2]
 => 2
[[4,5]]
 => [1,1]
 => [2]
 => 2
[[5,5]]
 => [2]
 => [1,1]
 => 1
[[1],[5]]
 => [1,1]
 => [2]
 => 2
[[2],[5]]
 => [1,1]
 => [2]
 => 2
[[3],[5]]
 => [1,1]
 => [2]
 => 2
[[4],[5]]
 => [1,1]
 => [2]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 9 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => 110 => 2
[[2,2]]
 => [2]
 => 100 => 1
[[1],[2]]
 => [1,1]
 => 110 => 2
[[1,3]]
 => [1,1]
 => 110 => 2
[[2,3]]
 => [1,1]
 => 110 => 2
[[3,3]]
 => [2]
 => 100 => 1
[[1],[3]]
 => [1,1]
 => 110 => 2
[[2],[3]]
 => [1,1]
 => 110 => 2
[[1,1,2]]
 => [2,1]
 => 1010 => 2
[[1,2,2]]
 => [2,1]
 => 1010 => 2
[[2,2,2]]
 => [3]
 => 1000 => 1
[[1,1],[2]]
 => [2,1]
 => 1010 => 2
[[1,2],[2]]
 => [2,1]
 => 1010 => 2
[[1,4]]
 => [1,1]
 => 110 => 2
[[2,4]]
 => [1,1]
 => 110 => 2
[[3,4]]
 => [1,1]
 => 110 => 2
[[4,4]]
 => [2]
 => 100 => 1
[[1],[4]]
 => [1,1]
 => 110 => 2
[[2],[4]]
 => [1,1]
 => 110 => 2
[[3],[4]]
 => [1,1]
 => 110 => 2
[[1,1,3]]
 => [2,1]
 => 1010 => 2
[[1,2,3]]
 => [1,1,1]
 => 1110 => 3
[[1,3,3]]
 => [2,1]
 => 1010 => 2
[[2,2,3]]
 => [2,1]
 => 1010 => 2
[[2,3,3]]
 => [2,1]
 => 1010 => 2
[[3,3,3]]
 => [3]
 => 1000 => 1
[[1,1],[3]]
 => [2,1]
 => 1010 => 2
[[1,2],[3]]
 => [1,1,1]
 => 1110 => 3
[[1,3],[2]]
 => [1,1,1]
 => 1110 => 3
[[1,3],[3]]
 => [2,1]
 => 1010 => 2
[[2,2],[3]]
 => [2,1]
 => 1010 => 2
[[2,3],[3]]
 => [2,1]
 => 1010 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 3
[[1,1,1,2]]
 => [3,1]
 => 10010 => 2
[[1,1,2,2]]
 => [2,2]
 => 1100 => 2
[[1,2,2,2]]
 => [3,1]
 => 10010 => 2
[[2,2,2,2]]
 => [4]
 => 10000 => 1
[[1,1,1],[2]]
 => [3,1]
 => 10010 => 2
[[1,1,2],[2]]
 => [2,2]
 => 1100 => 2
[[1,2,2],[2]]
 => [3,1]
 => 10010 => 2
[[1,1],[2,2]]
 => [2,2]
 => 1100 => 2
[[1,5]]
 => [1,1]
 => 110 => 2
[[2,5]]
 => [1,1]
 => 110 => 2
[[3,5]]
 => [1,1]
 => 110 => 2
[[4,5]]
 => [1,1]
 => 110 => 2
[[5,5]]
 => [2]
 => 100 => 1
[[1],[5]]
 => [1,1]
 => 110 => 2
[[2],[5]]
 => [1,1]
 => 110 => 2
[[3],[5]]
 => [1,1]
 => 110 => 2
[[4],[5]]
 => [1,1]
 => 110 => 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [2]
 => 2
[[2,2]]
 => [2]
 => [1,1]
 => 1
[[1],[2]]
 => [1,1]
 => [2]
 => 2
[[1,3]]
 => [1,1]
 => [2]
 => 2
[[2,3]]
 => [1,1]
 => [2]
 => 2
[[3,3]]
 => [2]
 => [1,1]
 => 1
[[1],[3]]
 => [1,1]
 => [2]
 => 2
[[2],[3]]
 => [1,1]
 => [2]
 => 2
[[1,1,2]]
 => [2,1]
 => [3]
 => 2
[[1,2,2]]
 => [2,1]
 => [3]
 => 2
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[2]]
 => [2,1]
 => [3]
 => 2
[[1,2],[2]]
 => [2,1]
 => [3]
 => 2
[[1,4]]
 => [1,1]
 => [2]
 => 2
[[2,4]]
 => [1,1]
 => [2]
 => 2
[[3,4]]
 => [1,1]
 => [2]
 => 2
[[4,4]]
 => [2]
 => [1,1]
 => 1
[[1],[4]]
 => [1,1]
 => [2]
 => 2
[[2],[4]]
 => [1,1]
 => [2]
 => 2
[[3],[4]]
 => [1,1]
 => [2]
 => 2
[[1,1,3]]
 => [2,1]
 => [3]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [2,1]
 => 3
[[1,3,3]]
 => [2,1]
 => [3]
 => 2
[[2,2,3]]
 => [2,1]
 => [3]
 => 2
[[2,3,3]]
 => [2,1]
 => [3]
 => 2
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 1
[[1,1],[3]]
 => [2,1]
 => [3]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [2,1]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [2,1]
 => 3
[[1,3],[3]]
 => [2,1]
 => [3]
 => 2
[[2,2],[3]]
 => [2,1]
 => [3]
 => 2
[[2,3],[3]]
 => [2,1]
 => [3]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [2,1]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [4]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [4]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [4]
 => 2
[[1,5]]
 => [1,1]
 => [2]
 => 2
[[2,5]]
 => [1,1]
 => [2]
 => 2
[[3,5]]
 => [1,1]
 => [2]
 => 2
[[4,5]]
 => [1,1]
 => [2]
 => 2
[[5,5]]
 => [2]
 => [1,1]
 => 1
[[1],[5]]
 => [1,1]
 => [2]
 => 2
[[2],[5]]
 => [1,1]
 => [2]
 => 2
[[3],[5]]
 => [1,1]
 => [2]
 => 2
[[4],[5]]
 => [1,1]
 => [2]
 => 2
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000733
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,2]]
 => [2]
 => [[1,2]]
 => 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => 2
[[3,3]]
 => [2]
 => [[1,2]]
 => 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => 1
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[4,4]]
 => [2]
 => [[1,2]]
 => 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => 1
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[5,5]]
 => [2]
 => [[1,2]]
 => 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => 2
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,2]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3,3]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[4,4]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[5,5]]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000007
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2,2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,3]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[4,4]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[5,5]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2,2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,3]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[1,1,2]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1,2,2]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
[[1,1],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1,2],[2]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[4,4]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[1,1,3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[[1,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[2,2,3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[2,3,3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
[[1,1],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[[1,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[2,2],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[2,3],[3]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[5,5]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000734
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2,2]]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
[[1],[2]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[1,3]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2,3]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[3,3]]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
[[1],[3]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2],[3]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[1,1,2]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2,2]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2,2]]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[[1,1],[2]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2],[2]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2,4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[3,4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[4,4]]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
[[1],[4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2],[4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[3],[4]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[1,1,3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[[1,3,3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2,3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,3,3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[3,3,3]]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[[1,1],[3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[[1,3],[3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,2],[3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[2,3],[3]]
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2,5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[3,5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[4,5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[5,5]]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
[[1],[5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[2],[5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[3],[5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
[[4],[5]]
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000757
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2,2]]
 => [2]
 => [[1,2]]
 => [2] => 1
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[3,3]]
 => [2]
 => [[1,2]]
 => [2] => 1
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[1,1,2]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1,2,2]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => [3] => 1
[[1,1],[2]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1,2],[2]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[4,4]]
 => [2]
 => [[1,2]]
 => [2] => 1
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[1,1,3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[[1,3,3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[2,2,3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[2,3,3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => [3] => 1
[[1,1],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[[1,3],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[2,2],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[2,3],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 2
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[[1,1,1,2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[[1,2,2,2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => [4] => 1
[[1,1,1],[2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[[1,2,2],[2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[5,5]]
 => [2]
 => [[1,2]]
 => [2] => 1
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 2
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000765The number of weak records in an integer composition. St001462The number of factors of a standard tableaux under concatenation. St000546The number of global descents of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001777The number of weak descents in an integer composition. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000809The reduced reflection length of the permutation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St001530The depth of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001401The number of distinct entries in a semistandard tableau. St000264The girth of a graph, which is not a tree. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph. 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. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000259The diameter of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.