Identifier
Values
[1,0] => [2,1] => [2,1] => ([(0,1)],2) => 2
[1,0,1,0] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3) => 2
[1,1,0,0] => [2,3,1] => [1,3,2] => ([(1,2)],3) => 2
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,0,0,1,0] => [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4) => 2
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,4,3] => ([(2,3)],4) => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,5,4] => ([(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => ([(4,5)],6) => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,1,4,5,6,7,8,2] => [3,1,4,5,6,8,2,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8) => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,4,1,5,6,7,8,3] => [4,1,2,5,6,8,3,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8) => 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,3,5,1,6,7,8,4] => [5,1,2,3,6,8,4,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8) => 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [2,3,4,6,1,7,8,5] => [6,1,2,3,4,8,5,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8) => 2
[] => [1] => [1] => ([],1) => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.