Identifier
Values
[1] => [[1]] => [1] => ([],1) => 0
[2] => [[1,2]] => [1,2] => ([],2) => 0
[1,1] => [[1],[2]] => [2,1] => ([(0,1)],2) => 2
[3] => [[1,2,3]] => [1,2,3] => ([],3) => 0
[2,1] => [[1,3],[2]] => [2,1,3] => ([(1,2)],3) => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 3
[4] => [[1,2,3,4]] => [1,2,3,4] => ([],4) => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => ([(2,3)],4) => 2
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => ([],5) => 0
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => ([(3,4)],5) => 2
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 4
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 3
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [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) => 5
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => ([],6) => 0
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => ([(4,5)],6) => 2
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 4
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 3
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 6
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [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) => 6
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => ([],7) => 0
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 2
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => 4
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 3
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 6
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => ([(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) => 6
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [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) => 7
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Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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.