- FindStat

Identifier

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001624: Lattices ⟶ ℤ

Values

[1] => ([],1) => ([],1) => 1
[1,2] => ([],2) => ([],1) => 1
[2,1] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,3] => ([],3) => ([],1) => 1
[1,3,2] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,1,3] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,3,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 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
[1,2,3,4] => ([],4) => ([],1) => 1
[1,2,4,3] => ([(2,3)],4) => ([(0,1)],2) => 1
[1,3,2,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[2,1,3,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,3,5,4] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,2,4,3,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,3,2,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[2,1,3,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,4,6,5] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,3,5,4,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,4,3,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,3,2,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[2,1,3,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,4,5,7,6] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,2,3,4,6,5,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,3,5,4,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,2,4,3,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,5,3,4,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[1,3,2,4,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,2,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,3,4,2,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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
[2,1,3,4,5,6,7] => ([(5,6)],7) => ([(0,1)],2) => 1
[2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[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

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 3\ q + 3\ q^{2}$

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

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.