- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,1,4]] => [1,2,3] => 00

[[1,2,4]] => [1,2,3] => 00

[[1,3,4]] => [1,2,3] => 00

[[1,4,4]] => [1,2,3] => 00

[[2,2,4]] => [1,2,3] => 00

[[2,3,4]] => [1,2,3] => 00

[[2,4,4]] => [1,2,3] => 00

[[3,3,4]] => [1,2,3] => 00

[[3,4,4]] => [1,2,3] => 00

[[4,4,4]] => [1,2,3] => 00

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

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

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

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

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

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

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

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

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

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

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

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

[[1],[2],[4]] => [3,2,1] => 11

[[1],[3],[4]] => [3,2,1] => 11

[[2],[3],[4]] => [3,2,1] => 11

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2507 entries. <<<

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

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.