Identifier
Values
[[1],[2],[3]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[2],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[3],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[3],[4]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1,2],[2],[3]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[3]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1],[2],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[3],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[3],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[4],[5]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1,2],[2],[4]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[4]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[3]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[3],[4]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,3],[3],[4]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,4],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[3],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[3],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[4],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[4],[5],[6]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1,2],[2],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[3]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[3],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[3],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[3],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[4],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,3],[3],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,4],[3],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[3],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,4],[4],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,4],[4],[5]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,5],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1],[2],[3],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[2],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,1],[2],[3],[4]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[2],[3],[4]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[3],[4]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2],[3],[4]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,1],[2,2],[3,3]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1],[2],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[3],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[3],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[2],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[4],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[3],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[4],[5],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[4],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[5],[6],[7]] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1,2],[2],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[2],[3]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[2],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[2],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[2],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[2],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[2],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[3],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[3],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,4],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[4],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,5],[5],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,6],[5],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,3],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,4],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[3],[4]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[3],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[3],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[3],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
>>> Load all 164 entries. <<<
[[2,4],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[4],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,5],[5],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[2,6],[5],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,4],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,5],[4],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,6],[4],[5]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,6],[4],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,5],[5],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[3,6],[5],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[4,5],[5],[6]] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4) => 2
[[4,6],[5],[6]] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1],[2],[3],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[3],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[3],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[2],[3],[4],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[2],[3],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[2],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[3],[4],[5],[6]] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,1],[2],[3],[5]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,1],[2],[4],[5]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,1],[3],[4],[5]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[2],[3],[5]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[2],[4],[5]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[3],[5]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,5],[2],[3],[5]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2],[4],[5]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,5],[2],[4],[5]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[3],[4],[5]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[3],[4],[5]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,5],[3],[4],[5]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[2,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[2,3],[3],[4],[5]] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[2,4],[3],[4],[5]] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[2,5],[3],[4],[5]] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,1],[2,2],[3,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,1],[2,2],[4,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,1],[2,3],[4,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,1],[3,3],[4,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[2,3],[4,4]] => [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[3,3],[4,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[2,2],[3,3],[4,4]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,1],[2,2],[3],[4]] => [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,2,1,1] => ([(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) => 2
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.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
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
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.