- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

[[1]] => [1] => [1] => 0

[[1,2]] => [1,2] => [2] => 0

[[1],[2]] => [2,1] => [1,1] => 1

[[1,2,3]] => [1,2,3] => [3] => 0

[[1,3],[2]] => [2,1,3] => [1,2] => 1

[[1,2],[3]] => [3,1,2] => [1,2] => 1

[[1],[2],[3]] => [3,2,1] => [1,1,1] => 3

[[1,2,3,4]] => [1,2,3,4] => [4] => 0

[[1,3,4],[2]] => [2,1,3,4] => [1,3] => 1

[[1,2,4],[3]] => [3,1,2,4] => [1,3] => 1

[[1,2,3],[4]] => [4,1,2,3] => [1,3] => 1

[[1,3],[2,4]] => [2,4,1,3] => [2,2] => 2

[[1,2],[3,4]] => [3,4,1,2] => [2,2] => 2

[[1,4],[2],[3]] => [3,2,1,4] => [1,1,2] => 3

[[1,3],[2],[4]] => [4,2,1,3] => [1,1,2] => 3

[[1,2],[3],[4]] => [4,3,1,2] => [1,1,2] => 3

[[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1] => 6

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

[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,4] => 1

[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,4] => 1

[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,4] => 1

[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,4] => 1

[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,3] => 2

[[1,2,5],[3,4]] => [3,4,1,2,5] => [2,3] => 2

[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,3] => 2

[[1,2,4],[3,5]] => [3,5,1,2,4] => [2,3] => 2

[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,3] => 2

[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,1,3] => 3

[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,1,3] => 3

[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,1,3] => 3

[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,1,3] => 3

[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,1,3] => 3

[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,1,3] => 3

[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,2,2] => 4

[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,2,2] => 4

[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,2] => 4

[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,2,2] => 4

[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,2] => 4

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

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

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

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

[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,1,1,1,1] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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.