- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤ (values match St000015The number of peaks of a Dyck path., St000053The number of valleys of the Dyck path., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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.