Identifier
Values
[1] => [[1]] => [1] => ([],1) => 1
[2] => [[1,2]] => [1,2] => ([],2) => 0
[1,1] => [[1],[2]] => [2,1] => ([(0,1)],2) => 1
[3] => [[1,2,3]] => [1,2,3] => ([],3) => 0
[2,1] => [[1,2],[3]] => [3,1,2] => ([(0,2),(1,2)],3) => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[4] => [[1,2,3,4]] => [1,2,3,4] => ([],4) => 0
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 3
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => ([],5) => 0
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 7
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 14
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 18
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 24
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => ([],6) => 0
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 15
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 31
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 46
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => ([(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
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 64
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 78
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => ([(0,2),(0,3),(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) => 96
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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) => 120
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => ([],7) => 0
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 31
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 32
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 115
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 146
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 162
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7) => 230
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 284
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 330
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 384
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 426
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 504
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 600
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 720
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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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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.