- FindStat

Identifier

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

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

2-conjugate

Description

Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.