Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
[[1,2]] => [1,2] => [1] => ([],1) => 0
[[1],[2]] => [2,1] => [1] => ([],1) => 0
[[1,2,3]] => [1,2,3] => [1,2] => ([],2) => 0
[[1,3],[2]] => [2,1,3] => [2,1] => ([(0,1)],2) => 1
[[1,2],[3]] => [3,1,2] => [1,2] => ([],2) => 0
[[1],[2],[3]] => [3,2,1] => [2,1] => ([(0,1)],2) => 1
[[1,2,3,4]] => [1,2,3,4] => [1,2,3] => ([],3) => 0
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3] => ([(1,2)],3) => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3] => ([],3) => 0
[[1,3],[2,4]] => [2,4,1,3] => [2,1,3] => ([(1,2)],3) => 1
[[1,4],[2],[3]] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[[1,3],[2],[4]] => [4,2,1,3] => [2,1,3] => ([(1,2)],3) => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4] => ([],4) => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4) => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4] => ([],4) => 0
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5) => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5) => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5) => 0
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5) => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5) => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [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) => 4
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [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) => 4
[[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([],6) => 0
[[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
[[1,2,3,4,6,7],[5]] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => 0
[[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,3,4,5,6],[2,7]] => [2,7,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
[[1,2,3,4,6],[5,7]] => [5,7,1,2,3,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,6,7],[4],[5]] => [5,4,1,2,3,6,7] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3,4,5,6],[2],[7]] => [7,2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => [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) => 3
[[1,2,5,6],[3,4,7]] => [3,4,7,1,2,5,6] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,4,5,6],[2,7],[3]] => [3,2,7,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,6],[4,7],[5]] => [5,4,7,1,2,3,6] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,5,6],[3,4],[7]] => [7,3,4,1,2,5,6] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,4,5,6],[2],[3],[7]] => [7,3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,6],[4],[5],[7]] => [7,5,4,1,2,3,6] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [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) => 3
[[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => [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) => 4
[[1,5,6],[2,7],[3],[4]] => [4,3,2,7,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [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) => 4
[[1,5,6],[2],[3],[4],[7]] => [7,4,3,2,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [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) => 4
[[1,6],[2,7],[3],[4],[5]] => [5,4,3,2,7,1,6] => [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) => 4
[[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [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) => 5
[[1,6],[2],[3],[4],[5],[7]] => [7,5,4,3,2,1,6] => [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) => 4
[[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [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) => 5
[[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([],7) => 0
[[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
[[1,2,3,4,6,7,8],[5]] => [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => 0
[[1,2,5,6,7,8],[3,4]] => [3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7) => 2
[[1,3,4,5,6,7],[2,8]] => [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
[[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,3,4,5,6,7],[2],[8]] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 1
[[1,2,3,7,8],[4,5,6]] => [4,5,6,1,2,3,7,8] => [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) => 3
[[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,4,5,6,7],[2],[3],[8]] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,7],[4,5,6,8]] => [4,5,6,8,1,2,3,7] => [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) => 3
[[1,2,7,8],[3,4],[5,6]] => [5,6,3,4,1,2,7,8] => [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) => 4
[[1,5,6,7],[2,8],[3],[4]] => [4,3,2,8,1,5,6,7] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [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) => 4
[[1,5,6,7],[2],[3],[4],[8]] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[1,2,7],[3,4],[5,6],[8]] => [8,5,6,3,4,1,2,7] => [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) => 4
[[1,6,7],[2,8],[3],[4],[5]] => [5,4,3,2,8,1,6,7] => [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) => 4
[[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [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) => 5
[[1,7],[2,8],[3],[4],[5],[6]] => [6,5,4,3,2,8,1,7] => [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) => 5
[[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [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) => 6
[[1,7],[2],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,1,7] => [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) => 5
[[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [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) => 6
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Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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.
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.
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
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
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