Your data matches 98 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001401
(load all 6 compositions to match this statistic)
Mapping
St001401: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 2
[[2,2]]
 => 1
[[1],[2]]
 => 2
[[1,1,2]]
 => 2
[[1,2,2]]
 => 2
[[2,2,2]]
 => 1
[[1,1],[2]]
 => 2
[[1,2],[2]]
 => 2
[[1,1,3]]
 => 2
[[1,2,3]]
 => 3
[[1,3,3]]
 => 2
[[2,2,3]]
 => 2
[[2,3,3]]
 => 2
[[3,3,3]]
 => 1
[[1,1],[3]]
 => 2
[[1,2],[3]]
 => 3
[[1,3],[2]]
 => 3
[[1,3],[3]]
 => 2
[[2,2],[3]]
 => 2
[[2,3],[3]]
 => 2
[[1],[2],[3]]
 => 3
[[1,1,1,2]]
 => 2
[[1,1,2,2]]
 => 2
[[1,2,2,2]]
 => 2
[[2,2,2,2]]
 => 1
[[1,1,1],[2]]
 => 2
[[1,1,2],[2]]
 => 2
[[1,2,2],[2]]
 => 2
[[1,1],[2,2]]
 => 2
[[1,1,1,3]]
 => 2
[[1,1,2,3]]
 => 3
[[1,1,3,3]]
 => 2
[[1,2,2,3]]
 => 3
[[1,2,3,3]]
 => 3
[[1,3,3,3]]
 => 2
[[2,2,2,3]]
 => 2
[[2,2,3,3]]
 => 2
[[2,3,3,3]]
 => 2
[[3,3,3,3]]
 => 1
[[1,1,1],[3]]
 => 2
[[1,1,2],[3]]
 => 3
[[1,1,3],[2]]
 => 3
[[1,1,3],[3]]
 => 2
[[1,2,2],[3]]
 => 3
[[1,2,3],[2]]
 => 3
[[1,2,3],[3]]
 => 3
[[1,3,3],[2]]
 => 3
[[1,3,3],[3]]
 => 2
[[2,2,2],[3]]
 => 2
[[2,2,3],[3]]
 => 2
Description
The number of distinct entries in a semistandard tableau.
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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => 2
[[1,1,2,3]]
 => [2,1,1]
 => 3
[[1,1,3,3]]
 => [2,2]
 => 2
[[1,2,2,3]]
 => [2,1,1]
 => 3
[[1,2,3,3]]
 => [2,1,1]
 => 3
[[1,3,3,3]]
 => [3,1]
 => 2
[[2,2,2,3]]
 => [3,1]
 => 2
[[2,2,3,3]]
 => [2,2]
 => 2
[[2,3,3,3]]
 => [3,1]
 => 2
[[3,3,3,3]]
 => [4]
 => 1
[[1,1,1],[3]]
 => [3,1]
 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => 3
[[1,1,3],[3]]
 => [2,2]
 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => 3
[[1,3,3],[3]]
 => [3,1]
 => 2
[[2,2,2],[3]]
 => [3,1]
 => 2
[[2,2,3],[3]]
 => [2,2]
 => 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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2,3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,1,3,3]]
 => [2,2]
 => [2,2]
 => 2
[[1,2,2,3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,2,3,3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,3,3]]
 => [2,2]
 => [2,2]
 => 2
[[2,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[3,3,3,3]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1,1],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,1,3],[3]]
 => [2,2]
 => [2,2]
 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => [3,1]
 => 3
[[1,3,3],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,3],[3]]
 => [2,2]
 => [2,2]
 => 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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => 10010 => 2
[[1,1,2,3]]
 => [2,1,1]
 => 10110 => 3
[[1,1,3,3]]
 => [2,2]
 => 1100 => 2
[[1,2,2,3]]
 => [2,1,1]
 => 10110 => 3
[[1,2,3,3]]
 => [2,1,1]
 => 10110 => 3
[[1,3,3,3]]
 => [3,1]
 => 10010 => 2
[[2,2,2,3]]
 => [3,1]
 => 10010 => 2
[[2,2,3,3]]
 => [2,2]
 => 1100 => 2
[[2,3,3,3]]
 => [3,1]
 => 10010 => 2
[[3,3,3,3]]
 => [4]
 => 10000 => 1
[[1,1,1],[3]]
 => [3,1]
 => 10010 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => 10110 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => 10110 => 3
[[1,1,3],[3]]
 => [2,2]
 => 1100 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => 10110 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => 10110 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => 10110 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => 10110 => 3
[[1,3,3],[3]]
 => [3,1]
 => 10010 => 2
[[2,2,2],[3]]
 => [3,1]
 => 10010 => 2
[[2,2,3],[3]]
 => [2,2]
 => 1100 => 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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2,3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,1,3,3]]
 => [2,2]
 => [4]
 => 2
[[1,2,2,3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,2,3,3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,3,3]]
 => [2,2]
 => [4]
 => 2
[[2,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 2
[[3,3,3,3]]
 => [4]
 => [1,1,1,1]
 => 1
[[1,1,1],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,1,3],[3]]
 => [2,2]
 => [4]
 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => [2,2]
 => 3
[[1,3,3],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2],[3]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,3],[3]]
 => [2,2]
 => [4]
 => 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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1,2,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,1,3,3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,2,3,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,3,3,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2,2,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2,3,3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[2,3,3,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[3,3,3,3]]
 => [4]
 => [[1,2,3,4]]
 => 1
[[1,1,1],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,1,3],[3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[[1,3,3],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2,2],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[2,2,3],[3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001227
(load all 4 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[[2,2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[[1,1,2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2,2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[2,2,2]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,1,3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2,3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[[1,3,3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[2,2,3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[2,3,3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[3,3,3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[[1,3],[2]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[[1,3],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[2,2],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[2,3],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[[1,1,1,2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,2,2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2,2,2]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,1,1],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,2],[2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,2],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[1,1,1,3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,2,3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,1,3,3]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,2,3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,2,3,3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,3,3,3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2,2,3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2,3,3]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[2,3,3,3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[3,3,3,3]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,1,1],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,1,3],[2]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,1,3],[3]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,2,3],[2]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,2,3],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,3,3],[2]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[1,3,3],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2,2],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[2,2,3],[3]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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
[[1,1,2]]
 => [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
[[1,1,3]]
 => [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
[[1,1,1,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1,2,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,1,3,3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[1,2,2,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,2,3,3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,3,3,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2,2,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2,3,3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[2,3,3,3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[3,3,3,3]]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,1,1],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[1,1,2],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,1,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,1,3],[3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[[1,2,2],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,2,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,2,3],[3]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,3,3],[2]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[[1,3,3],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2,2],[3]]
 => [3,1]
 => [[1,2,3],[4]]
 => 1 = 2 - 1
[[2,2,3],[3]]
 => [2,2]
 => [[1,2],[3,4]]
 => 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: St000319
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,2]]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2]]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1,2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2,2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[[1,1],[2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,1,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2,3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,3,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[[1,1],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2],[3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3],[2]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2,2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,1,1,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3,3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,3,3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[2,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[3,3,3,3]]
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[[1,1,1],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3],[3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,3],[3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,2]]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2]]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1,2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2,2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2,2]]
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[[1,1],[2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,1,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2,3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,3,3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[3,3,3]]
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[[1,1],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1,2],[3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3],[2]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,3],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,2],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => 2 = 3 - 1
[[1,1,1,2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2,2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2,2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2,2]]
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[[1,1,1],[2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,1,1,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3,3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3,3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,3,3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[2,3,3,3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[3,3,3,3]]
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[[1,1,1],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[1,1,2],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,1,3],[3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[[1,2,2],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,2,3],[3]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3],[2]]
 => [2,1,1]
 => [3,1]
 => 2 = 3 - 1
[[1,3,3],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,2],[3]]
 => [3,1]
 => [2,1,1]
 => 1 = 2 - 1
[[2,2,3],[3]]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000519The largest length of a factor maximising the subword complexity. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000653The last descent of a permutation. St000676The number of odd rises of 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. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001039The maximal height of a column in the parallelogram polyomino associated with 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001480The number of simple summands of the module J^2/J^3. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000446The disorder 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. St000546The number of global descents of a permutation. St000651The maximal size of a rise in a permutation. St000710The number of big deficiencies of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001489The maximum of the number of descents and the number of inverse descents. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001726The number of visible inversions of a permutation. St001777The number of weak descents in an integer composition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000454The largest eigenvalue of a graph if it is integral. 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. St000260The radius of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000259The diameter of a connected graph.