- FindStat

Identifier

Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤ

Values

[+] => [1] => ([],1) => 0

[-] => [1] => ([],1) => 0

[-,+] => [2,1] => ([(0,1)],2) => 1

[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3) => 1

[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3) => 1

[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 1

[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2

[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2

[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 1

[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2

[4,-,+,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => 2

[-,+,+,+,+] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1

[-,-,+,+,+] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

[-,+,-,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[-,+,+,-,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,-,-,+,+] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

[-,-,+,-,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[-,+,-,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,-,-,-,+] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1

[-,+,4,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,-,4,3,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[-,3,2,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[-,3,2,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,3,4,2,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,4,2,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[-,4,+,2,+] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2

[-,4,-,2,+] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2

[-,5,-,2,4] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[-,5,-,+,2] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2

[2,4,+,1,+] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[2,5,-,+,1] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[4,-,+,1,+] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2

[5,-,+,1,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

[5,-,+,+,1] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[5,+,-,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[5,-,-,+,1] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

[5,-,+,-,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[5,-,4,3,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[5,3,2,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

[-,+,+,+,+,+] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1

[-,-,+,+,+,+] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,+,-,+,+,+] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,+,+,-,+,+] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,+,+,-,+] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,-,-,+,+,+] => [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) => 2

[-,-,+,-,+,+] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,-,+,+,-,+] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,-,-,+,+] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,-,+,-,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,+,-,-,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,-,-,-,+,+] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,-,-,+,-,+] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,-,+,-,-,+] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,-,-,-,+] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,-,-,-,-,+] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1

[-,+,+,5,4,+] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,-,+,5,4,+] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,-,5,4,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,-,-,5,4,+] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,+,4,3,+,+] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,-,4,3,+,+] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,+,4,3,-,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,-,4,3,-,+] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,4,5,3,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,-,4,5,3,+] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,5,3,4,+] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,-,5,3,4,+] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,5,+,3,+] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,-,5,+,3,+] => [4,3,6,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,+,5,-,3,+] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,-,5,-,3,+] => [3,6,1,2,5,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 2

[-,+,6,-,3,5] => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2

[-,-,6,-,3,5] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 2

[-,+,6,-,+,3] => [2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 2

[-,-,6,-,+,3] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 2

[-,3,2,+,+,+] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,3,2,-,+,+] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,3,2,+,-,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,3,2,-,-,+] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,3,2,5,4,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,3,4,2,+,+] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,3,4,2,-,+] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,3,4,5,2,+] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,3,5,2,4,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,3,5,+,2,+] => [4,2,6,1,3,5] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 2

[-,3,5,-,2,+] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,3,6,-,2,5] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2

[-,3,6,-,+,2] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,4,2,3,+,+] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,4,2,3,-,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,4,2,5,3,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,4,+,2,+,+] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 2

[-,4,-,2,+,+] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

[-,4,+,2,-,+] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,4,-,2,-,+] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,4,+,5,2,+] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,4,-,5,2,+] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,4,5,2,3,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

[-,4,5,3,2,+] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2

[-,5,2,3,4,+] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

[-,5,2,+,3,+] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2

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$F_{1} = 2$

Description

The radius of a connected graph.
This is the minimum eccentricity of any vertex.

Map

lower permutation

Description

The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let

$I$ be the anti-exceedance set of a decorated permutation

$w$ . Let

$v$ be the

$k$ -Grassmannian permutation determined by

$v[k] = w^{-1}(I)$ and let

$u$ be the permutation satisfying

$u = wv$ . Then

$[u, v]$ is the Grassmann interval corresponding to

$w$ .
This map returns

$u$ .

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.