- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤ (values match St000932The number of occurrences of the pattern UDU in a Dyck path., St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 11 + 9\ q + 3\ q^{2} + q^{3}$

$F_{5} = 53 + 44\ q + 18\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 309 + 265\ q + 110\ q^{2} + 30\ q^{3} + 5\ q^{4} + q^{5}$

Description

Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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.