Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [2,1] => ([(0,1)],2) => 1
{{1},{2}} => [1,2] => [1,2] => ([],2) => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
{{1,2},{3}} => [2,1,3] => [2,1,3] => ([(1,2)],3) => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => ([(1,2)],3) => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([],3) => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
{{1,2,4},{3}} => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4) => 1
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 1
{{1,3},{2,4}} => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
{{1,4},{2,3}} => [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}} => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4) => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4) => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([],4) => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [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}} => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => 1
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [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},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5) => 1
{{1,3,4,5},{2}} => [3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [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
{{1,3,5},{2,4}} => [3,4,5,2,1] => [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
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [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
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [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
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => [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}} => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [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},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [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
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [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
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,5},{2,4},{3}} => [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},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => 2
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [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},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5) => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [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},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5) => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([],5) => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [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
{{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => [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
{{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => [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}} => [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => [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},{3,4,6},{5}} => [2,1,4,6,5,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => 1
{{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => 1
{{1,2,5},{3},{4,6}} => [2,5,3,6,1,4] => [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
{{1,2,5},{3},{4},{6}} => [2,5,3,4,1,6] => [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
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [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},{3,5},{4,6}} => [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3,5},{4},{6}} => [2,1,5,4,3,6] => [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2},{3,6},{4,5}} => [2,1,6,5,4,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => 1
{{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3},{4,6},{5}} => [2,1,3,6,5,4] => [2,1,3,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
{{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => 1
{{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6) => 1
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => [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
{{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,3,4},{2,5},{6}} => [3,5,4,1,2,6] => [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
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => [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
{{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => [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
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => [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
{{1,3},{2,4},{5,6}} => [3,4,1,2,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => [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
{{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => [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
{{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => [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
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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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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