- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000872: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3$

$F_{4} = 4 + q$

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

Description

The number of very big descents of a permutation.
A very big descent of a permutation

$\pi$ is an index

$i$ such that

$\pi_i - \pi_{i+1} > 2$ .
For the number of descents, see St000021The number of descents of a permutation. and for the number of big descents, see St000647The number of big descents of a permutation.. General

$r$ -descents were for example be studied in [1, Section 2].

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

Map

to Dyck path

Description

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

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.