- FindStat

Identifier

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Map

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

Description

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

Map

reverse

Description

The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.