- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [1,1,1,1] => [1,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1418 entries. <<<

search for individual values

searching the database for the individual values of this statistic

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

first row removal

Description

Removes the first entry of an integer partition

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition

$\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition

$(n-1,\ldots,2,1)$ when cutting out

$D$ considered as a path from

$(0,0)$ to

$(n,n)$ .
In other words,

$\lambda_{i}$ is the number of down-steps before the

$(n+1-i)$ -th up-step of

$D$ .
This map is a bijection between Dyck paths of size

$n$ and partitions inside the staircase partition

$(n-1,\ldots,2,1)$ .