- FindStat

Identifier

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤ (values match St000099The number of valleys of a permutation, including the boundary.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

[8,1] => [4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8,2] => [4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => 2

[8,1,1] => [4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8,3] => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 1

[8,2,1] => [4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 2

[8,1,1,1] => [4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 131 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

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q$

$F_{4} = 2 + 3\ q$

$F_{5} = 3 + 4\ q$

Description

The number of inner peaks of a permutation.
The number of peaks including the boundary is St000092The number of outer peaks of a permutation..

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

Glaisher-Franklin

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.