- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤ (values match St000023The number of inner peaks of a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 131 entries. <<<

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Description

The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is St000353The number of inner valleys 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

to Dyck path

Description

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