- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2465 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 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.
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 maxima 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 weak deficiency positions of $\chi(\pi)$, and
the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.