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Your data matches 47 different statistics following compositions of up to 3 maps.
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Matching statistic: St000384
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Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000384: 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],[2]]
=> [1,1]
=> 2
[[1,3]]
=> [1,1]
=> 2
[[2,3]]
=> [1,1]
=> 2
[[3,3]]
=> [2]
=> 2
[[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]
=> 3
[[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]
=> 2
[[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]
=> 3
[[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]
=> 3
[[1,1,2,2]]
=> [2,2]
=> 3
[[1,2,2,2]]
=> [3,1]
=> 3
[[2,2,2,2]]
=> [4]
=> 4
[[1,1,1],[2]]
=> [3,1]
=> 3
[[1,1,2],[2]]
=> [2,2]
=> 3
[[1,2,2],[2]]
=> [3,1]
=> 3
[[1,1],[2,2]]
=> [2,2]
=> 3
[[1,5]]
=> [1,1]
=> 2
[[2,5]]
=> [1,1]
=> 2
[[3,5]]
=> [1,1]
=> 2
[[4,5]]
=> [1,1]
=> 2
[[5,5]]
=> [2]
=> 2
[[1],[5]]
=> [1,1]
=> 2
[[2],[5]]
=> [1,1]
=> 2
[[3],[5]]
=> [1,1]
=> 2
[[4],[5]]
=> [1,1]
=> 2
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000380
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Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,1]
=> 3 = 2 + 1
[[2,2]]
=> [2]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> 3 = 2 + 1
[[1,3]]
=> [1,1]
=> 3 = 2 + 1
[[2,3]]
=> [1,1]
=> 3 = 2 + 1
[[3,3]]
=> [2]
=> 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> 3 = 2 + 1
[[2,2,2]]
=> [3]
=> 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> 3 = 2 + 1
[[1,4]]
=> [1,1]
=> 3 = 2 + 1
[[2,4]]
=> [1,1]
=> 3 = 2 + 1
[[3,4]]
=> [1,1]
=> 3 = 2 + 1
[[4,4]]
=> [2]
=> 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> 3 = 2 + 1
[[3,3,3]]
=> [3]
=> 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> 4 = 3 + 1
[[1,5]]
=> [1,1]
=> 3 = 2 + 1
[[2,5]]
=> [1,1]
=> 3 = 2 + 1
[[3,5]]
=> [1,1]
=> 3 = 2 + 1
[[4,5]]
=> [1,1]
=> 3 = 2 + 1
[[5,5]]
=> [2]
=> 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> 3 = 2 + 1
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St000019
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Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ 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]
=> [2,3,1] => 2
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[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] => 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
[[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] => 3
[[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] => 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
[[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] => 3
[[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] => 3
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[[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] => 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
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000288
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —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]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000336
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ 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]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
Description
The leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Matching statistic: St000875
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000875: Binary words ⟶ 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]
=> 101100 => 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001004
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001004: Permutations ⟶ 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]
=> [2,1,3] => 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000395
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Values
[[1,2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001020
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,2,2,2,2,2,2]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[[3,3,3,3,3,3,3]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 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]
=> ? = 7 + 1
[[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]
=> ? = 7 + 1
[[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]
=> ? = 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]
=> ? = 7 + 1
[[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]
=> ? = 7 + 1
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 4 = 2 + 2
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[1,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2,3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[3,3]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 4 = 2 + 2
[[1],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2],[3]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[1,1,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1,2,2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[3,4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[4,4]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 4 = 2 + 2
[[1],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[3],[4]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[1,1,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1,2,3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[1,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,2,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,3,3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1,2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[1,3],[2]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 2 + 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[1,1,1,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[[1,1,2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[1,2,2,2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[[1,1,2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[[1,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[3,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[4,5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[5,5]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 4 = 2 + 2
[[1],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[2],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[3],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 2
[[4],[5]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 2 + 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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 2
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000144The pyramid weight of the Dyck path. St000501The size of the first part in the decomposition of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000924The number of topologically connected components of a perfect matching. St000209Maximum difference of elements in cycles. St001480The number of simple summands of the module J^2/J^3. St001958The degree of the polynomial interpolating the values of a permutation. St000673The number of non-fixed points of a permutation. St001468The smallest fixpoint of a permutation. St000528The height of a poset. St000863The length of the first row of the shifted shape of a permutation. St000080The rank of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000782The indicator function of whether a given perfect matching is an L & P matching. St001645The pebbling number of a connected graph. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001409The maximal entry of a semistandard tableau. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. 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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