- FindStat

Identifier

Mp00245: Signed permutations —standardize⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions

Images

[1] => [1] => {{1}} => {{1}}

[-1] => [1] => {{1}} => {{1}}

[1,2] => [1,2] => {{1},{2}} => {{1},{2}}

[1,-2] => [1,2] => {{1},{2}} => {{1},{2}}

[-1,2] => [2,1] => {{1,2}} => {{1,2}}

[-1,-2] => [2,1] => {{1,2}} => {{1,2}}

[2,1] => [2,1] => {{1,2}} => {{1,2}}

[2,-1] => [1,2] => {{1},{2}} => {{1},{2}}

[-2,1] => [2,1] => {{1,2}} => {{1,2}}

[-2,-1] => [1,2] => {{1},{2}} => {{1},{2}}

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

[1,-2,-3] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

[-1,2,3] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-1,2,-3] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-1,-2,3] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-1,-2,-3] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

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

[1,3,-2] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[1,-3,2] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

[1,-3,-2] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[-1,3,2] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-1,3,-2] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-1,-3,2] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-1,-3,-2] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

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

[2,1,-3] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[2,-1,3] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

[2,-1,-3] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

[-2,1,3] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-2,1,-3] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-2,-1,3] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[-2,-1,-3] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

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

[2,3,-1] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[2,-3,1] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[2,-3,-1] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[-2,3,1] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-2,3,-1] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[-2,-3,1] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-2,-3,-1] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

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

[3,1,-2] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[3,-1,2] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[3,-1,-2] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

[-3,1,2] => [3,1,2] => {{1,2,3}} => {{1,2,3}}

[-3,1,-2] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[-3,-1,2] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[-3,-1,-2] => [1,3,2] => {{1},{2,3}} => {{1},{2,3}}

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

[3,2,-1] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[3,-2,1] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[3,-2,-1] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

[-3,2,1] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}}

[-3,2,-1] => [2,1,3] => {{1,2},{3}} => {{1,2},{3}}

[-3,-2,1] => [2,3,1] => {{1,2,3}} => {{1,2,3}}

[-3,-2,-1] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}}

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

[1,2,3,-4] => [1,2,3,4] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}}

[1,2,-3,4] => [1,2,4,3] => {{1},{2},{3,4}} => {{1},{2},{3,4}}

[1,2,-3,-4] => [1,2,4,3] => {{1},{2},{3,4}} => {{1},{2},{3,4}}

[1,-2,3,4] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-2,3,-4] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-2,-3,4] => [1,4,3,2] => {{1},{2,4},{3}} => {{1},{2,4},{3}}

[1,-2,-3,-4] => [1,4,3,2] => {{1},{2,4},{3}} => {{1},{2,4},{3}}

[-1,2,3,4] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,2,3,-4] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,2,-3,4] => [4,1,3,2] => {{1,2,4},{3}} => {{1,2,4},{3}}

[-1,2,-3,-4] => [4,1,3,2] => {{1,2,4},{3}} => {{1,2,4},{3}}

[-1,-2,3,4] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,-2,3,-4] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,-2,-3,4] => [4,3,2,1] => {{1,4},{2,3}} => {{1,3},{2,4}}

[-1,-2,-3,-4] => [4,3,2,1] => {{1,4},{2,3}} => {{1,3},{2,4}}

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

[1,2,4,-3] => [1,2,3,4] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}}

[1,2,-4,3] => [1,2,4,3] => {{1},{2},{3,4}} => {{1},{2},{3,4}}

[1,2,-4,-3] => [1,2,3,4] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}}

[1,-2,4,3] => [1,4,3,2] => {{1},{2,4},{3}} => {{1},{2,4},{3}}

[1,-2,4,-3] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-2,-4,3] => [1,4,3,2] => {{1},{2,4},{3}} => {{1},{2,4},{3}}

[1,-2,-4,-3] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[-1,2,4,3] => [4,1,3,2] => {{1,2,4},{3}} => {{1,2,4},{3}}

[-1,2,4,-3] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,2,-4,3] => [4,1,3,2] => {{1,2,4},{3}} => {{1,2,4},{3}}

[-1,2,-4,-3] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,-2,4,3] => [4,3,2,1] => {{1,4},{2,3}} => {{1,3},{2,4}}

[-1,-2,4,-3] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}}

[-1,-2,-4,3] => [4,3,2,1] => {{1,4},{2,3}} => {{1,3},{2,4}}

[-1,-2,-4,-3] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}}

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

[1,3,2,-4] => [1,3,2,4] => {{1},{2,3},{4}} => {{1},{2,3},{4}}

[1,3,-2,4] => [1,2,4,3] => {{1},{2},{3,4}} => {{1},{2},{3,4}}

[1,3,-2,-4] => [1,2,4,3] => {{1},{2},{3,4}} => {{1},{2},{3,4}}

[1,-3,2,4] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-3,2,-4] => [1,4,2,3] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-3,-2,4] => [1,3,4,2] => {{1},{2,3,4}} => {{1},{2,3,4}}

[1,-3,-2,-4] => [1,3,4,2] => {{1},{2,3,4}} => {{1},{2,3,4}}

[-1,3,2,4] => [4,2,1,3] => {{1,3,4},{2}} => {{1,3,4},{2}}

[-1,3,2,-4] => [4,2,1,3] => {{1,3,4},{2}} => {{1,3,4},{2}}

[-1,3,-2,4] => [4,1,3,2] => {{1,2,4},{3}} => {{1,2,4},{3}}

>>> Load all 4282 entries. <<<

Map

standardize

Description

Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.

Map

to cycle type

Description

Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].

Map

Chen Deng Du Stanley Yan

Description

A map that swaps the crossing number and the nesting number of a set partition.