- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤ

Values

[[1,2]] => [1,2] => [1,2] => ([],2) => 1

[[2,2]] => [1,2] => [1,2] => ([],2) => 1

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

[[1,3]] => [1,2] => [1,2] => ([],2) => 1

[[2,3]] => [1,2] => [1,2] => ([],2) => 1

[[3,3]] => [1,2] => [1,2] => ([],2) => 1

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

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

[[1,1,2]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,2,2]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,2,2]] => [1,2,3] => [1,2,3] => ([],3) => 1

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

[[1,2],[2]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

[[1,4]] => [1,2] => [1,2] => ([],2) => 1

[[2,4]] => [1,2] => [1,2] => ([],2) => 1

[[3,4]] => [1,2] => [1,2] => ([],2) => 1

[[4,4]] => [1,2] => [1,2] => ([],2) => 1

[[1],[4]] => [2,1] => [2,1] => ([(0,1)],2) => 2

[[2],[4]] => [2,1] => [2,1] => ([(0,1)],2) => 2

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

[[1,1,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,2,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,3,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,2,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,3,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[3,3,3]] => [1,2,3] => [1,2,3] => ([],3) => 1

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

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

[[1,3],[2]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

[[1,3],[3]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

[[2,3],[3]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

[[1,1,1,2]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,1,2,2]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,2,2,2]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[2,2,2,2]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

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

[[1,1,2],[2]] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

[[1,2,2],[2]] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

[[1,5]] => [1,2] => [1,2] => ([],2) => 1

[[2,5]] => [1,2] => [1,2] => ([],2) => 1

[[3,5]] => [1,2] => [1,2] => ([],2) => 1

[[4,5]] => [1,2] => [1,2] => ([],2) => 1

[[5,5]] => [1,2] => [1,2] => ([],2) => 1

[[1],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 2

[[2],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 2

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

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

[[1,1,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,2,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,3,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[1,4,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,2,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,3,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[2,4,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[3,3,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[3,4,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

[[4,4,4]] => [1,2,3] => [1,2,3] => ([],3) => 1

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

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

[[1,4],[2]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

[[1,4],[3]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

[[1,4],[4]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

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

[[2,4],[3]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

[[2,4],[4]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

[[3,4],[4]] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

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

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

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

[[1,1,1,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,1,2,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,1,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,2,2,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,2,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[1,3,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[2,2,2,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[2,2,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[2,3,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[[3,3,3,3]] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

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

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

[[1,1,3],[2]] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

[[1,1,3],[3]] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

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

[[1,2,3],[2]] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

[[1,2,3],[3]] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

[[1,3,3],[2]] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

[[1,3,3],[3]] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

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

[[2,2,3],[3]] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

[[2,3,3],[3]] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

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

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

[[1,3],[2],[3]] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4) => 2

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

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

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$F_{(2, 2)} = 2\ q + q^{2}$

$F_{(2, 3)} = 3\ q + 2\ q^{2}$

$F_{(3, 2)} = 3\ q + 2\ q^{2}$

$F_{(2, 4)} = 4\ q + 3\ q^{2}$

$F_{(3, 3)} = 6\ q + 7\ q^{2}$

$F_{(2, 5)} = 5\ q + 4\ q^{2}$

$F_{(3, 4)} = 10\ q + 15\ q^{2}$

$F_{(2, 6)} = 6\ q + 5\ q^{2}$

$F_{(3, 5)} = 15\ q + 26\ q^{2}$

$F_{(2, 7)} = 7\ q + 6\ q^{2}$

$F_{(3, 6)} = 21\ q + 40\ q^{2}$

$F_{(2, 8)} = 8\ q + 7\ q^{2}$

$F_{(3, 7)} = 28\ q + 57\ q^{2}$

Description

The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost 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.

Map

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map

$\Phi$ in [1, sec.3]. In particular, it satisfies

$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.