Identifier
Values
[1] => [[1],[]] => ([],1) => ([],1) => 1
[2] => [[2],[]] => ([(0,1)],2) => ([],2) => 1
[1,1] => [[1,1],[]] => ([(0,1)],2) => ([],2) => 1
[3] => [[3],[]] => ([(0,2),(2,1)],3) => ([],3) => 1
[2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => ([(1,2)],3) => 2
[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => ([],3) => 1
[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
[3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 2
[2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
[2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 2
[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
[4,1] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 2
[3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 2
[2,1,1,1] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1
[5,1] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 2
[2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 2
[2,1,1,1,1] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1
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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
to skew partition
Description
The partition regarded as a skew partition.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
Map
incomparability graph
Description
The incomparability graph of a poset.