- FindStat

Identifier

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000800: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

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

$F_{5} = 10 + 17\ q + 4\ q^{2} + 2\ q^{3} + q^{4}$

Description

The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern

$(2,3,1)$ , such that the letter matched by

$2$ is the first entry of the permutation.

Map

Laplacian multiplicities

Description

The composition of multiplicities of the Laplacian eigenvalues.
Let

$\lambda_1 > \lambda_2 > \dots$ be the eigenvalues of the Laplacian matrix of a graph on

$n$ vertices. Then this map returns the composition

$a_1,\dots,a_k$ of

$n$ where

$a_i$ is the multiplicity of

$\lambda_i$ .

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

bounce path

Description

The bounce path determined by an integer composition.