- FindStat

Identifier

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000948: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

[2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 8

[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12

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

[3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 8

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

[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12

[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 14

[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 18

[4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12

[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12

[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 18

[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 18

[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 24

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

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

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

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

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

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

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

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

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

[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8

[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

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

[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8

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

[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 14

[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

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

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

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

[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

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

[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16

[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 16

[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8

[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 16

[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 28

[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36

[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 16

[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36

[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36

[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48

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

[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8

[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 16

[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 16

[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

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

[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36

[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 14

[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 28

[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36

[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 46

[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 54

[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 28

[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 42

>>> Load all 831 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

Description

The chromatic discriminant of a graph.
The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$.
According to [1], it equals the cardinality of any of the following sets:
(1) Acyclic orientations of G with unique sink at $q$,
(2) Maximum $G$-parking functions relative to $q$,
(3) Minimal $q$-critical states,
(4) Spanning trees of G without broken circuits,
(5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$,
(6) Multilinear Lyndon heaps on $G$.
In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.

Map

cone

Description

The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.

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.