Identifier
Values
[1] => [1] => ([],1) => ([],1) => 1
[1,2] => [2,1] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,1] => [1,2] => ([],2) => ([],1) => 1
[1,2,3] => [2,3,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,1,3] => [1,3,2] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,3,1] => [1,2,3] => ([],3) => ([],1) => 1
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1] => [2,1,3] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,1,4] => [1,2,4,3] => ([(2,3)],4) => ([(0,1)],2) => 1
[2,3,4,1] => [1,2,3,4] => ([],4) => ([],1) => 1
[2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,3,1] => [1,3,2,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,4,1] => [2,1,3,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[3,4,2,1] => [3,1,2,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,1,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,5,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,1,5] => [1,2,3,5,4] => ([(3,4)],5) => ([(0,1)],2) => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,3,5,1,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,5,4,1] => [1,2,4,3,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[2,4,3,1,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,3,5,1] => [1,3,2,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[2,4,5,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,3,4,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,4,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,2,4,1,5] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,4,5,1] => [2,1,3,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[3,2,5,4,1] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,4,2,5,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,3,5,1] => [2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,3,2,5,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,1,5,6] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,1,6,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,5,1,6] => [1,2,3,4,6,5] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,4,6,1,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,6,5,1] => [1,2,3,5,4,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,3,5,4,1,6] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,5,4,6,1] => [1,2,4,3,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,3,5,6,4,1] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,6,4,5,1] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,6,5,4,1] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,4,3,5,1,6] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,3,5,6,1] => [1,3,2,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,4,3,6,5,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,5,3,6,1] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,3,4,6,1] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,4,3,6,1] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,2,4,5,1,6] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,4,5,6,1] => [2,1,3,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[3,2,4,6,5,1] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,5,4,6,1] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,4,2,5,6,1] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,3,5,6,1] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,3,2,5,6,1] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,4,5,6,1,7] => [1,2,3,4,5,7,6] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,5,7,1,6] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,5,6,4,7,1] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[2,4,3,5,6,1,7] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,3,5,6,7,1] => [1,3,2,4,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,3,6,5,7,1] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,5,3,6,7,1] => [1,4,2,3,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,2,4,5,6,1,7] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,4,5,6,7,1] => [2,1,3,4,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,5,4,6,7,1] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,4,2,5,6,7,1] => [3,1,2,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,3,5,6,7,1] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,3,2,5,6,7,1] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
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 breadth of a lattice.
The breadth of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Map
connected vertex partitions
Description
Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
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.