Your data matches 141 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000358
(load all 4 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 1
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001727
(load all 18 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 0
Description
The number of invisible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000356
(load all 8 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,1,3] => 0
[2,1,3] => [1,3,2] => [2,1,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,3,4] => 0
[1,4,3,2] => [1,4,2,3] => [2,1,3,4] => 0
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 1
[2,1,4,3] => [1,4,2,3] => [2,1,3,4] => 0
[2,3,1,4] => [1,4,2,3] => [2,1,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 0
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => 0
[3,1,4,2] => [1,4,2,3] => [2,1,3,4] => 0
[3,2,1,4] => [1,4,2,3] => [2,1,3,4] => 0
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,4,3] => 1
[4,2,1,3] => [1,3,2,4] => [1,2,4,3] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,4,5] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,4,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,4,3,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,4,5,3] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,4,5,1,3] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,3,5] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,3,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,5,3,4] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,5,3,1,4] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,5,3,1,4] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,5,3,4] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => 2
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000039
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,4,3,5,2] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,3,5,4,2] => 1
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000497
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => ? = 0
[1,2] => [1,2] => {{1},{2}}
 => 0
[2,1] => [1,2] => {{1},{2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 0
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => 0
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 0
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => 0
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 0
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 0
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 0
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => 0
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => 0
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 0
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 0
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 0
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
Description
The lcb statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000491
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => {{1}}
 => ? = 0
[1,2] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 0
[2,1] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => {{1,2},{3}}
 => 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => {{1,2},{3}}
 => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000609
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => {{1}}
 => ? = 0
[1,2] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 0
[2,1] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 0
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 0
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 0
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 0
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 0
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 0
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 0
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 0
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
 => ? = 0 + 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
[2,1,3] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,2,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1} + 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[2,1,3,5,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,1,4,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[2,5,1,4,3] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[2,5,3,1,4] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[2,5,4,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,1,4,2,5] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,2,4,5,1] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,2,5,1,4] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,1,3,2,5] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,2,5,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,3,5,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[4,3,5,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2} + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000454
(load all 9 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => [1,2] => ([],2)
 => 0
[2,1] => [1,2] => [1,2] => ([],2)
 => 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
 => 0
[2,1,3] => [1,3,2] => [1,2,3] => ([],3)
 => 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
 => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => ([],4)
 => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[1,5,3,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,5,3] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,3,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[2,5,1,4,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[2,5,3,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,4,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,5,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,4,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,5,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000681
(load all 3 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
 => []
 => ? = 0
[1,2] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,1,3] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,3,1] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,2,4,3] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,3,2,4] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,3,4,2] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,2,3] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,3,4] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,4,3] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,1,4] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,1] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,3] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,3,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[3,1,2,4] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,2] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,1,4] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,4,1] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,1,2] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[4,1,2,3] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,3,2] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[4,2,1,3] => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,3,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[4,3,1,2] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,2,3,4,5] => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[1,4,2,3,5] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[1,5,2,3,4] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[3,2,1,5,4] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[3,4,1,5,2] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[3,4,2,1,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[3,4,2,5,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[3,4,5,2,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[3,5,1,4,2] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[3,5,2,1,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[3,5,2,4,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[4,1,2,5,3] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,1,3,2,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,1,3,5,2] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[4,2,1,5,3] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[4,2,3,1,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,2,3,5,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[4,5,1,3,2] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[4,5,2,1,3] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,5,2,3,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[4,5,3,1,2] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[5,1,2,4,3] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[5,1,3,2,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000567The sum of the products of all pairs of parts. St001964The interval resolution global dimension of a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001811The Castelnuovo-Mumford regularity of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St001621The number of atoms of a lattice. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001864The number of excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.