- FindStat

Identifier

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2466 entries. <<<

Map

catabolism

Description

Remove the first row of the semistandard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry

$k$ into tableau

$T$ is as follows:
If

$k$ is larger than all entries in the first column, place

$k$ at the bottom of the first column and the procedure is finished. Otherwise, place

$k$ in the first column, replacing the smallest entry,

$y$ , greater or equal to than

$k$ . Now insert

$y$ into the second column using the same procedure: if

$y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place

$y$ in the second column, replacing, or 'bumping', the smallest entry,

$z$ , larger than or equal to

$y$ . Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).

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 views to invisible inversion bottoms

Description

Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation

$\sigma$ is a pair

$i < j$ such that

$i < \sigma(j) < \sigma(i)$ . The element

$\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation

$\pi$ is an element

$\pi(j)$ such that

$\pi(i+1) < \pi(j) < \pi(i)$ , and additionally the smallest element in the decreasing run containing

$\pi(i)$ is smaller than the smallest element in the decreasing run containing

$\pi(j)$ .
This map is a bijection

$\chi:\mathfrak S_n \to \mathfrak S_n$ , such that

the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$ ,
the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$ ,
the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$ ,
the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$ .