Values
([],1) => ([],1) => ([],1) => 1
([],2) => ([],1) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => ([],2) => 2
([],3) => ([],1) => ([],1) => 1
([(1,2)],3) => ([(0,1)],2) => ([],2) => 2
([(0,2),(1,2)],3) => ([(0,1)],2) => ([],2) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([],4) => ([],1) => ([],1) => 1
([(2,3)],4) => ([(0,1)],2) => ([],2) => 2
([(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 2
([(0,3),(1,2)],4) => ([(0,1)],2) => ([],2) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => ([],2) => 2
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([],2) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 24
([],5) => ([],1) => ([],1) => 1
([(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(1,4),(2,3)],5) => ([(0,1)],2) => ([],2) => 2
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 20
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 24
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 24
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 24
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 24
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 120
([],6) => ([],1) => ([],1) => 1
([(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(2,5),(3,4)],6) => ([(0,1)],2) => ([],2) => 2
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => ([],2) => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => ([],2) => 2
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 2
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 20
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 6
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 20
>>> Load all 208 entries. <<<
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 number of connected components of the friends and strangers graph.
Let $X$ and $Y$ be graphs with the same vertex set $\{1,\dots,n\}$. Then the friends-and-strangers graph has as vertex set the set of permutations $\mathfrak S_n$ and edges $\left(\sigma, (i, j)\circ\sigma\right)$ if $(i, j)$ is an edge of $X$ and $\left(\sigma(i), \sigma(j)\right)$ is an edge of $Y$.
This statistic is the number of connected components of the friends and strangers graphs where $X=Y$.
For example, if $X$ is a complete graph the statistic is $1$, if $X$ has no edges, the statistic is $n!$, and if $X$ is the path graph, the statistic is
$$ \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k (n-k)!\binom{n-k}{k}, $$
see [thm. 2.2, 3].
Let $X$ and $Y$ be graphs with the same vertex set $\{1,\dots,n\}$. Then the friends-and-strangers graph has as vertex set the set of permutations $\mathfrak S_n$ and edges $\left(\sigma, (i, j)\circ\sigma\right)$ if $(i, j)$ is an edge of $X$ and $\left(\sigma(i), \sigma(j)\right)$ is an edge of $Y$.
This statistic is the number of connected components of the friends and strangers graphs where $X=Y$.
For example, if $X$ is a complete graph the statistic is $1$, if $X$ has no edges, the statistic is $n!$, and if $X$ is the path graph, the statistic is
$$ \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k (n-k)!\binom{n-k}{k}, $$
see [thm. 2.2, 3].
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
Map
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!