- FindStat

Identifier

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2\ q$

Description

The Colin de Verdière graph invariant.

Map

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

Description

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

Map

graph of inversions

Description

The graph of inversions of a permutation.
For a permutation of

$\{1,\dots,n\}$ , this is the graph with vertices

$\{1,\dots,n\}$ , where

$(i,j)$ is an edge if and only if it is an inversion of the permutation.

Map

reflect parallelogram polyomino

Description

Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.