- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

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

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

[[1,2,3]] => [1,2,3] => 00 => 0

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

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

[[1],[2],[3]] => [3,2,1] => 11 => 3

[[1,2,3,4]] => [1,2,3,4] => 000 => 0

[[1,3,4],[2]] => [2,1,3,4] => 100 => 1

[[1,2,4],[3]] => [3,1,2,4] => 100 => 1

[[1,2,3],[4]] => [4,1,2,3] => 100 => 1

[[1,3],[2,4]] => [2,4,1,3] => 010 => 2

[[1,2],[3,4]] => [3,4,1,2] => 010 => 2

[[1,4],[2],[3]] => [3,2,1,4] => 110 => 3

[[1,3],[2],[4]] => [4,2,1,3] => 110 => 3

[[1,2],[3],[4]] => [4,3,1,2] => 110 => 3

[[1],[2],[3],[4]] => [4,3,2,1] => 111 => 6

[[1,2,3,4,5]] => [1,2,3,4,5] => 0000 => 0

[[1,3,4,5],[2]] => [2,1,3,4,5] => 1000 => 1

[[1,2,4,5],[3]] => [3,1,2,4,5] => 1000 => 1

[[1,2,3,5],[4]] => [4,1,2,3,5] => 1000 => 1

[[1,2,3,4],[5]] => [5,1,2,3,4] => 1000 => 1

[[1,3,5],[2,4]] => [2,4,1,3,5] => 0100 => 2

[[1,2,5],[3,4]] => [3,4,1,2,5] => 0100 => 2

[[1,3,4],[2,5]] => [2,5,1,3,4] => 0100 => 2

[[1,2,4],[3,5]] => [3,5,1,2,4] => 0100 => 2

[[1,2,3],[4,5]] => [4,5,1,2,3] => 0100 => 2

[[1,4,5],[2],[3]] => [3,2,1,4,5] => 1100 => 3

[[1,3,5],[2],[4]] => [4,2,1,3,5] => 1100 => 3

[[1,2,5],[3],[4]] => [4,3,1,2,5] => 1100 => 3

[[1,3,4],[2],[5]] => [5,2,1,3,4] => 1100 => 3

[[1,2,4],[3],[5]] => [5,3,1,2,4] => 1100 => 3

[[1,2,3],[4],[5]] => [5,4,1,2,3] => 1100 => 3

[[1,4],[2,5],[3]] => [3,2,5,1,4] => 1010 => 4

[[1,3],[2,5],[4]] => [4,2,5,1,3] => 1010 => 4

[[1,2],[3,5],[4]] => [4,3,5,1,2] => 1010 => 4

[[1,3],[2,4],[5]] => [5,2,4,1,3] => 1010 => 4

[[1,2],[3,4],[5]] => [5,3,4,1,2] => 1010 => 4

[[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1110 => 6

[[1,4],[2],[3],[5]] => [5,3,2,1,4] => 1110 => 6

[[1,3],[2],[4],[5]] => [5,4,2,1,3] => 1110 => 6

[[1,2],[3],[4],[5]] => [5,4,3,1,2] => 1110 => 6

[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 1111 => 10

[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 00000 => 0

[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 10000 => 1

[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => 10000 => 1

[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => 10000 => 1

[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => 10000 => 1

[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 10000 => 1

[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => 01000 => 2

[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 01000 => 2

[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => 01000 => 2

[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => 01000 => 2

[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => 01000 => 2

[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => 01000 => 2

[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => 01000 => 2

[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => 01000 => 2

[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 01000 => 2

[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 11000 => 3

[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => 11000 => 3

[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => 11000 => 3

[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => 11000 => 3

[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => 11000 => 3

[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => 11000 => 3

[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => 11000 => 3

[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => 11000 => 3

[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => 11000 => 3

[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 11000 => 3

[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => 00100 => 3

[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => 00100 => 3

[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => 00100 => 3

[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => 00100 => 3

[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 00100 => 3

[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => 10100 => 4

[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 10100 => 4

[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => 10100 => 4

[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => 10100 => 4

[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => 10100 => 4

[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => 10100 => 4

[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => 10100 => 4

[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => 10100 => 4

[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => 10100 => 4

[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => 10100 => 4

[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => 10100 => 4

[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => 10100 => 4

[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => 10100 => 4

[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => 10100 => 4

[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => 10100 => 4

[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 10100 => 4

[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 11100 => 6

[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => 11100 => 6

[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => 11100 => 6

[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => 11100 => 6

[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => 11100 => 6

[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => 11100 => 6

[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => 11100 => 6

[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => 11100 => 6

[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => 11100 => 6

[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 11100 => 6

[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => 01010 => 6

[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => 01010 => 6

[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => 01010 => 6

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Description

The sum of the positions of the ones in a binary word.

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.

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.