Identifier
Values
[1,3,2] => [3,1,2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 4
[1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 4
[3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 4
[1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,5,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2,1,4,5] => [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) => 4
[3,4,1,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,3,1,2,5] => [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) => 4
[4,5,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,4,1,2,3] => [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) => 4
[1,3,2,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,5,2,3,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,6,2,3,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,3,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,4,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,5,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,6,1,3,4,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2,1,4,5,6] => [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) => 4
[3,4,1,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,5,1,2,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,6,1,2,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,3,1,2,5,6] => [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) => 4
[4,5,1,2,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,6,1,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,4,1,2,3,6] => [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) => 4
[5,6,1,2,3,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[6,5,1,2,3,4] => [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) => 4
[1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,6,1,3,4,5,7] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2,7,1,3,4,5,6] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2,1,4,5,6,7] => [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) => 4
[3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,6,1,2,4,5,7] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,7,1,2,4,5,6] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,3,1,2,5,6,7] => [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) => 4
[4,5,1,2,3,6,7] => [1,2,4,5,3,6,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,6,1,2,3,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[4,7,1,2,3,5,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,4,1,2,3,6,7] => [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) => 4
[5,6,1,2,3,4,7] => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,7,1,2,3,4,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[6,5,1,2,3,4,7] => [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) => 4
[6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[7,6,1,2,3,4,5] => [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) => 4
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules with projective dimension at most 1.
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.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).