- FindStat

Identifier

Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

[+] => [1] => [1] => [.,.]

[-] => [1] => [1] => [.,.]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[-,+,+,+,+] => [2,3,4,5,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]

[+,-,+,+,+] => [1,3,4,5,2] => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]

[+,+,-,+,+] => [1,2,4,5,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]

[+,+,+,-,+] => [1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]

[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]

[-,-,+,+,+] => [3,4,5,1,2] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]

[-,+,-,+,+] => [2,4,5,1,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]

[-,+,+,-,+] => [2,3,5,1,4] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]

[-,+,+,+,-] => [2,3,4,1,5] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]

[+,-,-,+,+] => [1,4,5,2,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]

[+,-,+,-,+] => [1,3,5,2,4] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]

[+,-,+,+,-] => [1,3,4,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]

>>> Load all 2371 entries. <<<

Map

upper permutation

Description

The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.

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.