Identifier
-
Mp00075:
Semistandard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤ
Values
[[1,2,2],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,2,3],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,3],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,3],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,3,3],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[3]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,2,2,2],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,4],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,4],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,4],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,4],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,4],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,4],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,3,4],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,4,4],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,4,4],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[3,4,4],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,4],[3],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[2,4],[3],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,2,2,3],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,3,3],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,3,3],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,3,3],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[2,3,3,3],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,3],[2,3]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,2,3],[2],[3]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,3,3],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,2,2,2,2],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,5],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,5],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,5],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,5,5],[2]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,3,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,5,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,4,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,5,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,5,5],[5]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,3,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,4,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,5,5],[3]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,4,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,5,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[2,5,5],[5]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[3,4,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[3,5,5],[4]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[3,5,5],[5]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[4,5,5],[5]] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[3]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,5],[3],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,5],[3],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,5],[4],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[2,5],[3],[4]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[2,5],[3],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[2,5],[4],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[3,5],[4],[5]] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2
[[1,2,2,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,3,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,4,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,3,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,4,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,4,4,4],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,3,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,4,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,4,4,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,4,4,4],[4]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[2,3,3,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[2,3,4,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[2,4,4,4],[3]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[2,4,4,4],[4]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[3,4,4,4],[4]] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,4],[2,3]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,2,4],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,3,4],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,3,4],[3,4]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[2,3,4],[3,4]] => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,2,4],[2],[3]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,2,4],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,3,4],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,3,4],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,4,4],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,4,4],[2],[4]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,3,4],[3],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,4,4],[3],[4]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[2,3,4],[3],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[2,4,4],[3],[4]] => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,3],[2,4],[3]] => [3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,4],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,2,2,3],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,2,3,3],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,3,3],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,3,3,3],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,3,3,3],[3]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[2,3,3,3,3],[3]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,2,3],[2,3]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[[1,2,3,3],[2,3]] => [2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[1,2,2,3],[2],[3]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
>>> Load all 331 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).
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.
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.
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