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
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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).