- FindStat

Your data matches 187 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000169
(load all 3 compositions to match this statistic)

Mapping

St000169: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 0
[[1,2]]
 => 0
[[1],[2]]
 => 1
[[1,2,3]]
 => 0
[[1,3],[2]]
 => 2
[[1,2],[3]]
 => 1
[[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 2
[[1,2,3],[4]]
 => 1
[[1,2],[3,4]]
 => 2
[[1,2,3,4,5]]
 => 0
[[1,3,4,5],[2]]
 => 4
[[1,2,4,5],[3]]
 => 3
[[1,2,3,5],[4]]
 => 2
[[1,2,3,4],[5]]
 => 1
[[1,2,5],[3,4]]
 => 3
[[1,2,3],[4,5]]
 => 2
[[1,2,3,4,5,6]]
 => 0
[[1,3,4,5,6],[2]]
 => 5
[[1,2,4,5,6],[3]]
 => 4
[[1,2,3,5,6],[4]]
 => 3
[[1,2,3,4,6],[5]]
 => 2
[[1,2,3,4,5],[6]]
 => 1
[[1,2,5,6],[3,4]]
 => 4
[[1,2,3,6],[4,5]]
 => 3
[[1,2,3,4],[5,6]]
 => 2
[[1,2,3],[4,5,6]]
 => 3
[[1,2,3,4,5,6,7]]
 => 0
[[1,3,4,5,6,7],[2]]
 => 6
[[1,2,4,5,6,7],[3]]
 => 5
[[1,2,3,5,6,7],[4]]
 => 4
[[1,2,3,4,6,7],[5]]
 => 3
[[1,2,3,4,5,7],[6]]
 => 2
[[1,2,3,4,5,6],[7]]
 => 1
[[1,2,5,6,7],[3,4]]
 => 5
[[1,2,3,6,7],[4,5]]
 => 4
[[1,2,3,4,7],[5,6]]
 => 3
[[1,2,3,4,5],[6,7]]
 => 2
[[1,2,3,7],[4,5,6]]
 => 4
[[1,2,3,4],[5,6,7]]
 => 3

Description

The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.

Matching statistic: St001697
(load all 3 compositions to match this statistic)

Mapping

St001697: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 0
[[1,2]]
 => 0
[[1],[2]]
 => 1
[[1,2,3]]
 => 0
[[1,3],[2]]
 => 2
[[1,2],[3]]
 => 1
[[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 2
[[1,2,3],[4]]
 => 1
[[1,2],[3,4]]
 => 2
[[1,2,3,4,5]]
 => 0
[[1,3,4,5],[2]]
 => 4
[[1,2,4,5],[3]]
 => 3
[[1,2,3,5],[4]]
 => 2
[[1,2,3,4],[5]]
 => 1
[[1,2,5],[3,4]]
 => 3
[[1,2,3],[4,5]]
 => 2
[[1,2,3,4,5,6]]
 => 0
[[1,3,4,5,6],[2]]
 => 5
[[1,2,4,5,6],[3]]
 => 4
[[1,2,3,5,6],[4]]
 => 3
[[1,2,3,4,6],[5]]
 => 2
[[1,2,3,4,5],[6]]
 => 1
[[1,2,5,6],[3,4]]
 => 4
[[1,2,3,6],[4,5]]
 => 3
[[1,2,3,4],[5,6]]
 => 2
[[1,2,3],[4,5,6]]
 => 3
[[1,2,3,4,5,6,7]]
 => 0
[[1,3,4,5,6,7],[2]]
 => 6
[[1,2,4,5,6,7],[3]]
 => 5
[[1,2,3,5,6,7],[4]]
 => 4
[[1,2,3,4,6,7],[5]]
 => 3
[[1,2,3,4,5,7],[6]]
 => 2
[[1,2,3,4,5,6],[7]]
 => 1
[[1,2,5,6,7],[3,4]]
 => 5
[[1,2,3,6,7],[4,5]]
 => 4
[[1,2,3,4,7],[5,6]]
 => 3
[[1,2,3,4,5],[6,7]]
 => 2
[[1,2,3,7],[4,5,6]]
 => 4
[[1,2,3,4],[5,6,7]]
 => 3

Description

The shifted natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.

Matching statistic: St000009

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => 0
[[1,2]]
 => [[1],[2]]
 => 0
[[1],[2]]
 => [[1,2]]
 => 1
[[1,2,3]]
 => [[1],[2],[3]]
 => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [[1,3],[2]]
 => 1
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 0
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 2
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 0
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 4
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 3
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 2
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 5
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => 4
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => 3
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => 2
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 1
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 4
[[1,2,3,6],[4,5]]
 => [[1,4],[2,5],[3],[6]]
 => 3
[[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => 2
[[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 3
[[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 0
[[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 6
[[1,2,4,5,6,7],[3]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => 5
[[1,2,3,5,6,7],[4]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => 4
[[1,2,3,4,6,7],[5]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => 3
[[1,2,3,4,5,7],[6]]
 => [[1,6],[2],[3],[4],[5],[7]]
 => 2
[[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 1
[[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,4],[2,5],[3],[6],[7]]
 => 4
[[1,2,3,4,7],[5,6]]
 => [[1,5],[2,6],[3],[4],[7]]
 => 3
[[1,2,3,4,5],[6,7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => 2
[[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 4
[[1,2,3,4],[5,6,7]]
 => [[1,5],[2,6],[3,7],[4]]
 => 3

Description

The charge of a standard tableau.

Matching statistic: St000330

Mapping

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => 0
[[1,2]]
 => [[1,2]]
 => 0
[[1],[2]]
 => [[1],[2]]
 => 1
[[1,2,3]]
 => [[1,2,3]]
 => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [[1,3],[2]]
 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 3
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 2
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 1
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => 2
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 0
[[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => 4
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 3
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 2
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => 3
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => 2
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => 0
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => 5
[[1,2,4,5,6],[3]]
 => [[1,2,3,4,6],[5]]
 => 4
[[1,2,3,5,6],[4]]
 => [[1,2,3,5,6],[4]]
 => 3
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => 2
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => 4
[[1,2,3,6],[4,5]]
 => [[1,2,3,6],[4,5]]
 => 3
[[1,2,3,4],[5,6]]
 => [[1,2,5,6],[3,4]]
 => 2
[[1,2,3],[4,5,6]]
 => [[1,2,3],[4,5,6]]
 => 3
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => 0
[[1,3,4,5,6,7],[2]]
 => [[1,2,3,4,5,6],[7]]
 => 6
[[1,2,4,5,6,7],[3]]
 => [[1,2,3,4,5,7],[6]]
 => 5
[[1,2,3,5,6,7],[4]]
 => [[1,2,3,4,6,7],[5]]
 => 4
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,5,6,7],[4]]
 => 3
[[1,2,3,4,5,7],[6]]
 => [[1,2,4,5,6,7],[3]]
 => 2
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => 1
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,5],[6,7]]
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,7],[5,6]]
 => 4
[[1,2,3,4,7],[5,6]]
 => [[1,2,3,6,7],[4,5]]
 => 3
[[1,2,3,4,5],[6,7]]
 => [[1,2,5,6,7],[3,4]]
 => 2
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4],[5,6,7]]
 => 4
[[1,2,3,4],[5,6,7]]
 => [[1,2,3,7],[4,5,6]]
 => 3

Description

The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

Matching statistic: St000738

Mapping

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => 1 = 0 + 1
[[1,2]]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[2]]
 => [[1],[2]]
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => 1 = 0 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => 3 = 2 + 1
[[1,2],[3]]
 => [[1,3],[2]]
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => 5 = 4 + 1
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 3 = 2 + 1
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => 6 = 5 + 1
[[1,2,4,5,6],[3]]
 => [[1,2,3,4,6],[5]]
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => [[1,2,3,5,6],[4]]
 => 4 = 3 + 1
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => 3 = 2 + 1
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => 2 = 1 + 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => 5 = 4 + 1
[[1,2,3,6],[4,5]]
 => [[1,2,3,6],[4,5]]
 => 4 = 3 + 1
[[1,2,3,4],[5,6]]
 => [[1,2,5,6],[3,4]]
 => 3 = 2 + 1
[[1,2,3],[4,5,6]]
 => [[1,2,3],[4,5,6]]
 => 4 = 3 + 1
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => 1 = 0 + 1
[[1,3,4,5,6,7],[2]]
 => [[1,2,3,4,5,6],[7]]
 => 7 = 6 + 1
[[1,2,4,5,6,7],[3]]
 => [[1,2,3,4,5,7],[6]]
 => 6 = 5 + 1
[[1,2,3,5,6,7],[4]]
 => [[1,2,3,4,6,7],[5]]
 => 5 = 4 + 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,5,6,7],[4]]
 => 4 = 3 + 1
[[1,2,3,4,5,7],[6]]
 => [[1,2,4,5,6,7],[3]]
 => 3 = 2 + 1
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => 2 = 1 + 1
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,5],[6,7]]
 => 6 = 5 + 1
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,7],[5,6]]
 => 5 = 4 + 1
[[1,2,3,4,7],[5,6]]
 => [[1,2,3,6,7],[4,5]]
 => 4 = 3 + 1
[[1,2,3,4,5],[6,7]]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 2 + 1
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4],[5,6,7]]
 => 5 = 4 + 1
[[1,2,3,4],[5,6,7]]
 => [[1,2,3,7],[4,5,6]]
 => 4 = 3 + 1

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000008

Mapping

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [1] => 0
[[1,2]]
 => [[1,2]]
 => [2] => 0
[[1],[2]]
 => [[1],[2]]
 => [1,1] => 1
[[1,2,3]]
 => [[1,2,3]]
 => [3] => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => [2,1] => 2
[[1,2],[3]]
 => [[1,3],[2]]
 => [1,2] => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [4] => 0
[[1,3,4],[2]]
 => [[1,2,3],[4]]
 => [3,1] => 3
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => [2,2] => 2
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => [1,3] => 1
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => [2,2] => 2
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [5] => 0
[[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => [4,1] => 4
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => 3
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => [2,3] => 2
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => [3,2] => 3
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [6] => 0
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 5
[[1,2,4,5,6],[3]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => 4
[[1,2,3,5,6],[4]]
 => [[1,2,3,5,6],[4]]
 => [3,3] => 3
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => [2,4] => 2
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 4
[[1,2,3,6],[4,5]]
 => [[1,2,3,6],[4,5]]
 => [3,3] => 3
[[1,2,3,4],[5,6]]
 => [[1,2,5,6],[3,4]]
 => [2,4] => 2
[[1,2,3],[4,5,6]]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 3
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [7] => 0
[[1,3,4,5,6,7],[2]]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 6
[[1,2,4,5,6,7],[3]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => 5
[[1,2,3,5,6,7],[4]]
 => [[1,2,3,4,6,7],[5]]
 => [4,3] => 4
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,5,6,7],[4]]
 => [3,4] => 3
[[1,2,3,4,5,7],[6]]
 => [[1,2,4,5,6,7],[3]]
 => [2,5] => 2
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => 1
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,7],[5,6]]
 => [4,3] => 4
[[1,2,3,4,7],[5,6]]
 => [[1,2,3,6,7],[4,5]]
 => [3,4] => 3
[[1,2,3,4,5],[6,7]]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => 2
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 4
[[1,2,3,4],[5,6,7]]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => 3

Description

The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St001232
(load all 5 compositions to match this statistic)

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1,0,0]
 => 0
[[1],[2]]
 => [1,1] => [1,0,1,0]
 => 1
[[1,2,3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1,3],[2]]
 => [1,2] => [1,0,1,1,0,0]
 => 2
[[1,2],[3]]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[[1,2,3,4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,3,4],[2]]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
[[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,3],[4]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,2],[3,4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,4,5],[2]]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,4,5],[3]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4],[5]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,2,5],[3,4]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3],[4,5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6],[2]]
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,4,5,6],[3]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,5,6],[4]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5],[6]]
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,2,5,6],[3,4]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,6],[4,5]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3],[4,5,6]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6,7],[2]]
 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[[1,2,4,5,6,7],[3]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,3,5,6,7],[4]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,4,6,7],[5]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5,6],[7]]
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[[1,2,5,6,7],[3,4]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,3,6,7],[4,5]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,4,7],[5,6]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,7],[4,5,6]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,4],[5,6,7]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St000734

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => [[1]]
 => 1 = 0 + 1
[[1,2]]
 => [[1],[2]]
 => [[1],[2]]
 => 1 = 0 + 1
[[1],[2]]
 => [[1,2]]
 => [[1,2]]
 => 2 = 1 + 1
[[1,2,3]]
 => [[1],[2],[3]]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 3 = 2 + 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 3 = 2 + 1
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [[1,5],[2],[3],[4]]
 => 5 = 4 + 1
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => [[1,4],[2],[3],[5]]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => 3 = 2 + 1
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [[1,4],[2,5],[3]]
 => 4 = 3 + 1
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 6 = 5 + 1
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => [[1,5],[2],[3],[4],[6]]
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => [[1,4],[2],[3],[5],[6]]
 => 4 = 3 + 1
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [[1,3],[2],[4],[5],[6]]
 => 3 = 2 + 1
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => 2 = 1 + 1
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => [[1,5],[2,6],[3],[4]]
 => 5 = 4 + 1
[[1,2,3,6],[4,5]]
 => [[1,4],[2,5],[3],[6]]
 => [[1,4],[2,5],[3],[6]]
 => 4 = 3 + 1
[[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => [[1,3],[2,4],[5],[6]]
 => 3 = 2 + 1
[[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => [[1,4],[2,5],[3,6]]
 => 4 = 3 + 1
[[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1 = 0 + 1
[[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 7 = 6 + 1
[[1,2,4,5,6,7],[3]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => [[1,6],[2],[3],[4],[5],[7]]
 => 6 = 5 + 1
[[1,2,3,5,6,7],[4]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => 5 = 4 + 1
[[1,2,3,4,6,7],[5]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => 4 = 3 + 1
[[1,2,3,4,5,7],[6]]
 => [[1,6],[2],[3],[4],[5],[7]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => 3 = 2 + 1
[[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 2 = 1 + 1
[[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => 6 = 5 + 1
[[1,2,3,6,7],[4,5]]
 => [[1,4],[2,5],[3],[6],[7]]
 => [[1,5],[2,6],[3],[4],[7]]
 => 5 = 4 + 1
[[1,2,3,4,7],[5,6]]
 => [[1,5],[2,6],[3],[4],[7]]
 => [[1,4],[2,5],[3],[6],[7]]
 => 4 = 3 + 1
[[1,2,3,4,5],[6,7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 3 = 2 + 1
[[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => [[1,5],[2,6],[3,7],[4]]
 => 5 = 4 + 1
[[1,2,3,4],[5,6,7]]
 => [[1,5],[2,6],[3,7],[4]]
 => [[1,4],[2,5],[3,6],[7]]
 => 4 = 3 + 1

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000839
(load all 6 compositions to match this statistic)

Mapping

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [[1]]
 => {{1}}
 => 1 = 0 + 1
[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 1 = 0 + 1
[[1],[2]]
 => [[1],[2]]
 => {{1},{2}}
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 1 = 0 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 3 = 2 + 1
[[1,2],[3]]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 3 = 2 + 1
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 5 = 4 + 1
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3 = 2 + 1
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => 2 = 1 + 1
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4 = 3 + 1
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 3 = 2 + 1
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 6 = 5 + 1
[[1,2,4,5,6],[3]]
 => [[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => [[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => 4 = 3 + 1
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => {{1,2,4,5,6},{3}}
 => 3 = 2 + 1
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => 2 = 1 + 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 5 = 4 + 1
[[1,2,3,6],[4,5]]
 => [[1,2,3,6],[4,5]]
 => {{1,2,3,6},{4,5}}
 => 4 = 3 + 1
[[1,2,3,4],[5,6]]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 3 = 2 + 1
[[1,2,3],[4,5,6]]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 4 = 3 + 1
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => {{1,2,3,4,5,6,7}}
 => 1 = 0 + 1
[[1,3,4,5,6,7],[2]]
 => [[1,2,3,4,5,6],[7]]
 => {{1,2,3,4,5,6},{7}}
 => 7 = 6 + 1
[[1,2,4,5,6,7],[3]]
 => [[1,2,3,4,5,7],[6]]
 => {{1,2,3,4,5,7},{6}}
 => 6 = 5 + 1
[[1,2,3,5,6,7],[4]]
 => [[1,2,3,4,6,7],[5]]
 => {{1,2,3,4,6,7},{5}}
 => 5 = 4 + 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,5,6,7],[4]]
 => {{1,2,3,5,6,7},{4}}
 => 4 = 3 + 1
[[1,2,3,4,5,7],[6]]
 => [[1,2,4,5,6,7],[3]]
 => {{1,2,4,5,6,7},{3}}
 => 3 = 2 + 1
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => {{1,3,4,5,6,7},{2}}
 => 2 = 1 + 1
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,5],[6,7]]
 => {{1,2,3,4,5},{6,7}}
 => 6 = 5 + 1
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,7],[5,6]]
 => {{1,2,3,4,7},{5,6}}
 => 5 = 4 + 1
[[1,2,3,4,7],[5,6]]
 => [[1,2,3,6,7],[4,5]]
 => {{1,2,3,6,7},{4,5}}
 => 4 = 3 + 1
[[1,2,3,4,5],[6,7]]
 => [[1,2,5,6,7],[3,4]]
 => {{1,2,5,6,7},{3,4}}
 => 3 = 2 + 1
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4],[5,6,7]]
 => {{1,2,3,4},{5,6,7}}
 => 5 = 4 + 1
[[1,2,3,4],[5,6,7]]
 => [[1,2,3,7],[4,5,6]]
 => {{1,2,3,7},{4,5,6}}
 => 4 = 3 + 1

Description

The largest opener of a set partition. An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.

Matching statistic: St000010

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1,0]
 => []
 => 0
[[1,2]]
 => [2] => [1,1,0,0]
 => []
 => 0
[[1],[2]]
 => [1,1] => [1,0,1,0]
 => [1]
 => 1
[[1,2,3]]
 => [3] => [1,1,1,0,0,0]
 => []
 => 0
[[1,3],[2]]
 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
[[1,2],[3]]
 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
[[1,2,3,4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[[1,3,4],[2]]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[[1,2,3],[4]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[[1,2],[3,4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[[1,3,4,5],[2]]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[[1,2,4,5],[3]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[[1,2,3,4],[5]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[[1,2,5],[3,4]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[[1,2,3],[4,5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
[[1,3,4,5,6],[2]]
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[[1,2,4,5,6],[3]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
[[1,2,3,5,6],[4]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
[[1,2,3,4,5],[6]]
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 1
[[1,2,5,6],[3,4]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
[[1,2,3,6],[4,5]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
[[1,2,3],[4,5,6]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => 0
[[1,3,4,5,6,7],[2]]
 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => 6
[[1,2,4,5,6,7],[3]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => 5
[[1,2,3,5,6,7],[4]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[[1,2,3,4,6,7],[5]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3
[[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,5]
 => 2
[[1,2,3,4,5,6],[7]]
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => 1
[[1,2,5,6,7],[3,4]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => 5
[[1,2,3,6,7],[4,5]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[[1,2,3,4,7],[5,6]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3
[[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,5]
 => 2
[[1,2,3,7],[4,5,6]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[[1,2,3,4],[5,6,7]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3

Description

The length of the partition.

The following 177 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000081The number of edges of a graph. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000211The rank of the set partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000548The number of different non-empty partial sums of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000703The number of deficiencies of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001161The major index north count of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001721The degree of a binary word. St001826The maximal number of leaves on a vertex of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000451The length of the longest pattern of the form k 1 2. St000468The Hosoya index of a graph. St000505The biggest entry in the block containing the 1. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000693The modular (standard) major index of a standard tableau. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000391The sum of the positions of the ones in a binary word. St000446The disorder of a permutation. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000996The number of exclusive left-to-right maxima of a permutation. St000054The first entry of the permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000503The maximal difference between two elements in a common block. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001090The number of pop-stack-sorts needed to sort a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000833The comajor index of a permutation. St000653The last descent of a permutation. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000019The cardinality of the support of a permutation. St000662The staircase size of the code of a permutation. St000058The order of a permutation. St000956The maximal displacement of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001726The number of visible inversions of a permutation. St000470The number of runs in a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St000993The multiplicity of the largest part of an integer partition. St000007The number of saliances of the permutation. St000246The number of non-inversions of a permutation. St000651The maximal size of a rise in a permutation. St000209Maximum difference of elements in cycles. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001671Haglund's hag of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000501The size of the first part in the decomposition of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000304The load of a permutation. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000796The stat' of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000485The length of the longest cycle of a permutation. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in 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. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001346The number of parking functions that give the same permutation. St000739The first entry in the last row of a semistandard tableau. St000101The cocharge of a semistandard tableau. St000102The charge of a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St000456The monochromatic index of a connected graph. St000173The segment statistic of a semistandard tableau. St001209The pmaj statistic of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001330The hat guessing number of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.