Identifier
Values
[[1],[2]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[3]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[3]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[4]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[4]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[3],[4]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[2],[3]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[5]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[5]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[3],[5]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[4],[5]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[2],[4]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[3],[4]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[3],[4]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[6]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[6]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[3],[6]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[4],[6]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[5],[6]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[2],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[3],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[4],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[3],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[4],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[4],[5]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[3],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[4],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[5],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[6],[7]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[2],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[3],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[4],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[5],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[3],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[4],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[5],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[4],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[5],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[4],[5],[6]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[2],[3],[5]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[5]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[5]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[5]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[2],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[3],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[4],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[5],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[6],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[7],[8]] => [2,1] => [1,1] => ([(0,1)],2) => 2
[[1],[2],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[3],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[4],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[5],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[6],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[3],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[4],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[5],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[2],[6],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[4],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[5],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[3],[6],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[4],[5],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[4],[6],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[5],[6],[7]] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[[1],[2],[3],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[3],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[4],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[3],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[2],[4],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[3],[4],[5],[6]] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[[1]] => [1] => [1] => ([],1) => 1
[[2]] => [1] => [1] => ([],1) => 1
[[3]] => [1] => [1] => ([],1) => 1
[[4]] => [1] => [1] => ([],1) => 1
[[5]] => [1] => [1] => ([],1) => 1
[[6]] => [1] => [1] => ([],1) => 1
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(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) => 6
search for individual values
searching the database for the individual values of this statistic
Description
The pebbling number of a connected graph.
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.
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
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.