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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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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%
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
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(load all 10 compositions to match this statistic)
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%
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
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%
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
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(load all 3 compositions to match this statistic)
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%
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)
(load all 2 compositions to match this statistic)
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%
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: St000734
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%
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: St001227
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
[[1,2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,2,2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2,2]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,4]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2,3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,3,3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,3]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1,1,2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2,2,2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[5,5]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4,4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,2,4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,4,4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,4,4]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[2,2,2,2,2]]
=> [5]
=> []
=> []
=> ? = 1 - 1
[[6,6]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[5,5,5]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[4,4,4,4]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[3,3,3,3,3]]
=> [5]
=> []
=> []
=> ? = 1 - 1
[[2,2,2,2,2,2]]
=> [6]
=> []
=> []
=> ? = 1 - 1
[[7,7]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[6,6,6]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[5,5,5,5]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[4,4,4,4,4]]
=> [5]
=> []
=> []
=> ? = 1 - 1
[[3,3,3,3,3,3]]
=> [6]
=> []
=> []
=> ? = 1 - 1
[[2,2,2,2,2,2,2]]
=> [7]
=> []
=> []
=> ? = 1 - 1
[[8,8]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[7,7,7]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[6,6,6,6]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[5,5,5,5,5]]
=> [5]
=> []
=> []
=> ? = 1 - 1
[[4,4,4,4,4,4]]
=> [6]
=> []
=> []
=> ? = 1 - 1
[[3,3,3,3,3,3,3]]
=> [7]
=> []
=> []
=> ? = 1 - 1
[[2,2,2,2,2,2,2,2]]
=> [8]
=> []
=> []
=> ? = 1 - 1
[[1]]
=> [1]
=> []
=> []
=> ? = 1 - 1
[[2]]
=> [1]
=> []
=> []
=> ? = 1 - 1
[[1,1]]
=> [2]
=> []
=> []
=> ? = 1 - 1
[[3]]
=> [1]
=> []
=> []
=> ? = 1 - 1
[[1,1,1]]
=> [3]
=> []
=> []
=> ? = 1 - 1
[[4]]
=> [1]
=> []
=> []
=> ? = 1 - 1
[[1,1,1,1]]
=> [4]
=> []
=> []
=> ? = 1 - 1
[[5]]
=> [1]
=> []
=> []
=> ? = 1 - 1
[[1,1,1,1,1]]
=> [5]
=> []
=> []
=> ? = 1 - 1
[[6]]
=> [1]
=> []
=> []
=> ? = 1 - 1
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: St000653
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[[1,2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,2,2,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[3,3,3,3,3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,1,1,1,1,1,2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,2,2,2,2,2,2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 1
[[1,1,1,1,1,1],[2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,2,2,2,2,2],[2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[4,4,4,4,4,4]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,1,1,1,1,1,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,3,3,3,3,3,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[2,2,2,2,2,2,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[2,3,3,3,3,3,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 1
[[1,1,1,1,1,1],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,3,3,3,3,3],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[2,2,2,2,2,2],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[2,3,3,3,3,3],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? = 2
[[1,1,1,1,1,1,2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,1,2,2,2,2,2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? = 2
[[2,2,2,2,2,2,2,2]]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 1
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? = 2
[[1,1,1,1,1,1,2],[2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,1,2,2,2,2,2],[2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? = 2
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,2,3,4,5,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,4,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,4,6],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,5,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,5],[4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,5],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,4],[5,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4,5,6],[3]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4,5],[3,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4,5],[3],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,6],[4,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,3],[4,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St001480
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[[1,2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,2,2,2,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[3,3,3,3,3,3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1,1,1,1,2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,2,2,2,2,2,2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1,1,1,1],[2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,2,2,2,2,2],[2]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[4,4,4,4,4,4]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1,1,1,1,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,3,3,3,3,3,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,2,2,2,2,2,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,3,3,3,3,3,3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1,1,1,1],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,3,3,3,3,3],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,2,2,2,2,2],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,3,3,3,3,3],[3]]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1,1,1,1,1,1,2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1,1,1,1,1,2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,1,2,2,2,2,2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,2,2,2,2,2,2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[2,2,2,2,2,2,2,2]]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1,1,1,1,1],[2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1,1,1,1,1,2],[2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,1,2,2,2,2,2],[2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,2,2,2,2,2],[2]]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,4,5,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,6],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,5,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,5],[4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,5],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4],[5,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,5,6],[3]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,5],[3,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,5],[3],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,6],[4,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3],[4,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St000546The number of global descents of a permutation. St001777The number of weak descents in an integer composition. St000676The number of odd rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the 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. St000809The reduced reflection length of the permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001462The number of factors of a standard tableaux under concatenation. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima 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. 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. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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.
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