- FindStat

Identifier

Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[+,+,+,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,+,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,+,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,-,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,+,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,+,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,+,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,-,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,+,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,+,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,-,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,+,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,+,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,-,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,-,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,+,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,-,+,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,+,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,+,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,-,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,-,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,+,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,-,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,-,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,+,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,+,-,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,-,-,+] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,-,+,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,+,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,+,-,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[+,-,-,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

[-,-,-,-,-] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1743 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

LLPS

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.

Map

permutation

Description

The underlying permutation of the decorated permutation.