Identifier
Values
[1,0] => [1] => [1] => ([],1) => 1
[1,0,1,0] => [2,1] => [1,2] => ([],2) => 1
[1,1,0,0] => [1,2] => [2,1] => ([(0,1)],2) => 2
[1,0,1,0,1,0] => [2,3,1] => [1,2,3] => ([],3) => 1
[1,0,1,1,0,0] => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 3
[1,1,0,0,1,0] => [1,3,2] => [2,1,3] => ([(1,2)],3) => 2
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3) => 2
[1,1,1,0,0,0] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,2,3,4] => ([],4) => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 3
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4) => 2
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4) => 2
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 3
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,2,3,4,5] => ([],5) => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 3
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5) => 2
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5) => 2
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5) => 2
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5) => 2
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6) => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => [2,1,3,4,5,6] => ([(4,5)],6) => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,4,5,6,2] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [3,4,1,5,6,2] => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,0,1,0,0,1,0] => [3,4,5,1,6,2] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6) => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,1,2] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,2,3] => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,2,5,6,7,3,4] => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,2,3,6,7,4,5] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,1,2] => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,8,2,3] => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8) => 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [1,2,5,6,7,8,3,4] => [2,3,8,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0] => [1,2,3,6,7,8,4,5] => [2,3,4,8,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8) => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [1,2,3,4,7,8,5,6] => [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [1,2,3,4,5,8,6,7] => [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8) => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
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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
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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.