Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000039
(load all 7 compositions to match this statistic)
Mapping
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
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: St000223
(load all 5 compositions to match this statistic)
Mapping
Mp00239: Permutations —Corteel⟶ Permutations
St000223: 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] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => 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] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [3,2,4,1] => 1
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [2,4,3,1] => 1
[3,4,1,2] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [3,1,4,2] => 0
[4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [3,4,1,2] => 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] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,4,5,3] => 0
[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] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,5,2,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 2
Description
The number of nestings in the permutation.
Matching statistic: St000356
(load all 2 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: 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] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 2
[3,4,2,1] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[4,1,2,3] => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 0
[4,3,1,2] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 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: St000358
(load all 5 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: 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] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 0
[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] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 2
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,1,2] => 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[3,4,2,1] => [4,3,1,2] => [3,1,4,2] => 1
[4,1,2,3] => [2,3,4,1] => [4,3,2,1] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,2,1] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,3,1] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => [4,2,3,1] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[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] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,2,3] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => 2
[] => [] => [] => ? = 0
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: St001862
(load all 3 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 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] => 1
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [2,3,4,1] => 2
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,1,3,2] => 0
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 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] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => 0
[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] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,4,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,4,3,5,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2
[2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => ? = 1
[2,3,4,1,5] => [2,3,4,1,5] => ? = 2
[2,3,4,5,1] => [2,3,4,5,1] => ? = 3
[2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[2,4,3,5,1] => [2,4,3,5,1] => ? = 2
[2,4,5,1,3] => [2,4,5,1,3] => ? = 3
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,3,1,4] => [2,5,3,1,4] => ? = 1
[2,5,3,4,1] => [2,5,3,4,1] => ? = 1
[2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[2,5,4,3,1] => [2,5,4,3,1] => ? = 1
[3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[3,1,4,5,2] => [3,1,4,5,2] => ? = 2
[3,1,5,2,4] => [3,1,5,2,4] => ? = 1
[3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,2,5,4,1] => [3,2,5,4,1] => ? = 1
[3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[3,4,1,5,2] => [3,4,1,5,2] => ? = 3
[3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[3,4,5,1,2] => [3,4,5,1,2] => ? = 4
[3,4,5,2,1] => [3,4,5,2,1] => ? = 3
[3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[3,5,2,4,1] => [3,5,2,4,1] => ? = 1
[3,5,4,1,2] => [3,5,4,1,2] => ? = 3
[3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[4,1,2,5,3] => [4,1,2,5,3] => ? = 1
Description
The number of crossings of a signed permutation. A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that * $i < j \leq \pi(i) < \pi(j)$, or * $-i < j \leq -\pi(i) < \pi(j)$, or * $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001866
(load all 2 compositions to match this statistic)
Mapping
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [2,3,1] => 0
[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] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[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] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2
[2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
[2,3,5,1,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[2,3,5,4,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[2,4,1,5,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 2
[2,4,3,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[2,4,3,5,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[2,4,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 3
[2,4,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2
[2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[2,5,1,4,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[2,5,3,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[2,5,3,4,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[2,5,4,1,3] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 2
[2,5,4,3,1] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[3,1,4,5,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[3,1,5,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[3,1,5,4,2] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[3,2,4,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[3,2,4,5,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[3,2,5,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 1
[3,2,5,4,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[3,4,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1
[3,4,2,5,1] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 2
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[3,4,5,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[3,5,1,4,2] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[3,5,2,1,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[3,5,2,4,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[3,5,4,1,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 3
[3,5,4,2,1] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[4,1,2,5,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
Description
The nesting alignments of a signed permutation. A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that * $-i < -j < -\pi(j) < -\pi(i)$, or * $-i < j \leq \pi(j) < -\pi(i)$, or * $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
Mapping
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 + 3
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 0 + 3
[2,1] => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => ? = 0 + 3
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,3,1] => [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
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => ? = 0 + 3
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,3,1,4] => [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 = 1 + 3
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 3
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 3
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1 + 3
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 3
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 3
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 0 + 3
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,4,5,3] => [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
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,4,2,5] => [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 = 1 + 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 3
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 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 = 0 + 3
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 3
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1 + 3
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 3
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 3
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[2,3,1,4,5] => [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 = 1 + 3
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 3
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 3 + 3
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2 + 3
[2,3,5,4,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 2 + 3
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,3,5,6,4] => [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
[1,2,3,6,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,4,5,3,6] => [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 = 1 + 3
[1,2,5,3,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,4,2,5,6] => [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 = 1 + 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 = 0 + 3
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,3,1,4,5,6] => [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 = 1 + 3
[3,1,2,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
Description
The number of simple modules with projective dimension at most 1.