Identifier
Values
[[1]] => [1] => [1] => [[1],[]] => 0
[[1,2]] => [1,2] => [2] => [[2],[]] => 0
[[1],[2]] => [2,1] => [1,1] => [[1,1],[]] => 0
[[1,2,3]] => [1,2,3] => [3] => [[3],[]] => 0
[[1,3],[2]] => [2,1,3] => [1,2] => [[2,1],[]] => 0
[[1,2],[3]] => [3,1,2] => [1,2] => [[2,1],[]] => 0
[[1],[2],[3]] => [3,2,1] => [1,1,1] => [[1,1,1],[]] => 0
[[1,2,3,4]] => [1,2,3,4] => [4] => [[4],[]] => 0
[[1,3,4],[2]] => [2,1,3,4] => [1,3] => [[3,1],[]] => 0
[[1,2,4],[3]] => [3,1,2,4] => [1,3] => [[3,1],[]] => 0
[[1,2,3],[4]] => [4,1,2,3] => [1,3] => [[3,1],[]] => 0
[[1,3],[2,4]] => [2,4,1,3] => [2,2] => [[3,2],[1]] => 1
[[1,2],[3,4]] => [3,4,1,2] => [2,2] => [[3,2],[1]] => 1
[[1,4],[2],[3]] => [3,2,1,4] => [1,1,2] => [[2,1,1],[]] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [1,1,2] => [[2,1,1],[]] => 0
[[1,2],[3],[4]] => [4,3,1,2] => [1,1,2] => [[2,1,1],[]] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]] => 0
[[1,2,3,4,5]] => [1,2,3,4,5] => [5] => [[5],[]] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,4] => [[4,1],[]] => 0
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,4] => [[4,1],[]] => 0
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,4] => [[4,1],[]] => 0
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,4] => [[4,1],[]] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,3] => [[4,2],[1]] => 1
[[1,2,5],[3,4]] => [3,4,1,2,5] => [2,3] => [[4,2],[1]] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,3] => [[4,2],[1]] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [2,3] => [[4,2],[1]] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,3] => [[4,2],[1]] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,1,3] => [[3,1,1],[]] => 0
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,1,3] => [[3,1,1],[]] => 0
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,1,3] => [[3,1,1],[]] => 0
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,1,3] => [[3,1,1],[]] => 0
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,1,3] => [[3,1,1],[]] => 0
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,1,3] => [[3,1,1],[]] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,2,2] => [[3,2,1],[1]] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,2,2] => [[3,2,1],[1]] => 1
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,2] => [[3,2,1],[1]] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,2,2] => [[3,2,1],[1]] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,2] => [[3,2,1],[1]] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,1,1,2] => [[2,1,1,1],[]] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,1,1,2] => [[2,1,1,1],[]] => 0
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,1,1,2] => [[2,1,1,1],[]] => 0
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,1,1,2] => [[2,1,1,1],[]] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
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Description
The number of missing boxes of a skew partition.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.