Identifier
Values
[[1]] => [1] => [1] => ([],1) => 1
[[1],[2]] => [2,1] => [2,1] => ([(0,1)],2) => 2
[[1],[2],[3]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[[1,3],[2],[4]] => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => 4
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 5
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 5
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [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) => 2
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 6
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 6
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [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) => 2
[[1,3,4,6],[2],[5],[7]] => [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 7
[[1,4,6],[2],[3],[5],[7]] => [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 7
[[1,3,6],[2],[4],[5],[7]] => [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 7
[[1,4,5],[2],[3],[6],[7]] => [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[[1,3,5],[2],[4],[6],[7]] => [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 7
[[1,3,4],[2],[5],[6],[7]] => [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 7
[[1,6],[2],[3],[4],[5],[7]] => [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[1,5],[2],[3],[4],[6],[7]] => [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[1,4],[2],[3],[5],[6],[7]] => [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[1,3],[2],[4],[5],[6],[7]] => [7,6,5,4,2,1,3] => [6,1,7,5,4,3,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [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) => 2
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
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
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.