- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤ

Values

[1,0] => [1] => [1] => 0

[1,0,1,0] => [2,1] => [2] => 1

[1,1,0,0] => [1,2] => [1,1] => 0

[1,0,1,0,1,0] => [2,3,1] => [2,1] => 1

[1,0,1,1,0,0] => [2,1,3] => [2,1] => 1

[1,1,0,0,1,0] => [1,3,2] => [2,1] => 1

[1,1,0,1,0,0] => [3,1,2] => [2,1] => 1

[1,1,1,0,0,0] => [1,2,3] => [1,1,1] => 0

[1,0,1,0,1,0,1,0] => [2,3,4,1] => [2,1,1] => 1

[1,0,1,0,1,1,0,0] => [2,3,1,4] => [2,1,1] => 1

[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,2] => 2

[1,0,1,1,0,1,0,0] => [2,4,1,3] => [2,1,1] => 1

[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,1] => 1

[1,1,0,0,1,0,1,0] => [1,3,4,2] => [2,1,1] => 1

[1,1,0,0,1,1,0,0] => [1,3,2,4] => [2,1,1] => 1

[1,1,0,1,0,0,1,0] => [3,1,4,2] => [2,2] => 2

[1,1,0,1,0,1,0,0] => [3,4,1,2] => [2,1,1] => 1

[1,1,0,1,1,0,0,0] => [3,1,2,4] => [2,1,1] => 1

[1,1,1,0,0,0,1,0] => [1,2,4,3] => [2,1,1] => 1

[1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,1,1] => 1

[1,1,1,0,1,0,0,0] => [4,1,2,3] => [2,1,1] => 1

[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,1,1,1] => 0

[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [2,1,1,1] => 1

[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [2,1,1,1] => 1

[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [2,2,1] => 2

[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [2,1,1,1] => 1

[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [2,1,1,1] => 1

[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,2,1] => 2

[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,2,1] => 2

[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [2,2,1] => 2

[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [2,1,1,1] => 1

[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [2,1,1,1] => 1

[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,2,1] => 2

[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,2,1] => 2

[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [2,1,1,1] => 1

[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,1,1] => 1

[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [2,1,1,1] => 1

[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [2,1,1,1] => 1

[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [2,2,1] => 2

[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [2,1,1,1] => 1

[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [2,1,1,1] => 1

[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [2,2,1] => 2

[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [2,2,1] => 2

[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [2,2,1] => 2

[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [2,1,1,1] => 1

[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [2,1,1,1] => 1

[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [2,2,1] => 2

[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [2,2,1] => 2

[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [2,1,1,1] => 1

[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [2,1,1,1] => 1

[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [2,1,1,1] => 1

[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [2,1,1,1] => 1

[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [2,2,1] => 2

[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [2,1,1,1] => 1

[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [2,1,1,1] => 1

[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [2,2,1] => 2

[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [2,2,1] => 2

[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [2,1,1,1] => 1

[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [2,1,1,1] => 1

[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [2,1,1,1] => 1

[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [2,1,1,1] => 1

[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [2,1,1,1] => 1

[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [2,1,1,1] => 1

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,1,1,1,1] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [2,1,1,1,1] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [2,1,1,1,1] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [2,2,1,1] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [2,1,1,1,1] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [2,1,1,1,1] => 1

[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [2,2,1,1] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [2,2,1,1] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [2,2,1,1] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [2,1,1,1,1] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [2,1,1,1,1] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [2,2,1,1] => 2

[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [2,2,1,1] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [2,1,1,1,1] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [2,1,1,1,1] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,2,1,1] => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,2,1,1] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,2,2] => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,2,1,1] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,2,1,1] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [2,2,1,1] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [2,2,1,1] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [2,2,1,1] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [2,1,1,1,1] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [2,1,1,1,1] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [2,2,1,1] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [2,2,1,1] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [2,1,1,1,1] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [2,1,1,1,1] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,2,1,1] => 2

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,2,1,1] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,2,2] => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,2,1,1] => 2

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,2,1,1] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [2,2,1,1] => 2

[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [2,2,1,1] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [2,1,1,1,1] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [2,1,1,1,1] => 1

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Description

The number of even parts of a 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

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.