- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001934: Integer partitions ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [5] => 14

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5913 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$ (a_1, b_1),\dots,(a_r, b_r) $$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.