- FindStat

Identifier

Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = 6\ q + 11\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = 24\ q + 50\ q^{2} + 35\ q^{3} + 10\ q^{4} + q^{5}$

$F_{6} = 120\ q + 274\ q^{2} + 225\ q^{3} + 85\ q^{4} + 15\ q^{5} + q^{6}$

Description

The number of peaks of a Dyck path.

Map

inverse

Description

Sends a permutation to its inverse.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.