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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 1
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 1
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001557
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[4,3,2,1] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 1 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,3,4,5,6,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,3,4,6,2,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,3,5,2,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,3,5,6,4,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,3,6,2,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,3,6,5,2,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,2,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,2,6,3,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,4,5,3,6,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,5,6,2,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,6,3,2,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,4,6,5,3,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,2,3,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,2,6,4,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,5,4,2,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,4,6,3,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,6,2,3,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,5,6,3,4,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,2,5,3,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,3,4,5,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[1,6,4,2,3,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,4,5,2,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,5,2,4,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[1,6,5,3,2,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
[2,3,4,5,1,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,3,4,6,5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,3,5,1,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,3,5,4,6,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,3,6,1,5,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,3,6,4,1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,1,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,1,6,5,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,3,5,6,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,3,6,1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,5,3,1,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,4,6,3,5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
[2,5,1,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0 + 1
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001232
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,5,1,4,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[3,4,1,2,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,2,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,5,2,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,5,4,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[4,2,5,3,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[4,3,2,5,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 0
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,3,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,4,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,4,1,5] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 0
[2,3,5,4,1] => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[2,4,3,1,5] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,3,5,1] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,5,1,4,3] => [4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,5,3,1,4] => [3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,5,3,4,1] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,4,3,1] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,4,2,5] => [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,1,5,4,2] => [4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[3,2,1,4,5] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,2,1,5] => [4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,4,5,2,1] => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,1,2,4] => [3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,1,4,2] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,2,4,1] => [4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,5,4,1,2] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,1,5,2,3] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,2,5,1,3] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,2,5,3,1] => [2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,3,1,2,5] => [4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,3,2,1,5] => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,3,2,5,1] => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,3,5,1,2] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,1,3,2] => [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,2,1,3] => [4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,3,1,2] => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,3,2,1] => [3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,1,4,3,2] => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,2,4,1,3] => [2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,2,4,3,1] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,1,4,2] => [4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,3,2,1,4] => [3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,2,4,1] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,4,2,1] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,4,1,2,3] => [4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,4,2,3,1] => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001964
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Values
[1,2] => [2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,3] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,3,2,4] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[1,3,4,2] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[1,4,2,3] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[1,4,3,2] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,4,3] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[2,3,1,4] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,2,1,4] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,4,1] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,1,2] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,1,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,2,3,1] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,5,3,4] => [2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,5,4,3] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0
[1,3,4,5,2] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
[1,3,5,4,2] => [5,2,4,3,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,4,2,3,5] => [3,4,2,5,1] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0
[1,4,3,2,5] => [4,3,2,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0
[1,4,3,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,4,5,2,3] => [4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[1,4,5,3,2] => [5,4,2,3,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0
[1,5,2,3,4] => [3,4,5,2,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
[1,5,2,4,3] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0
[1,5,3,2,4] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0
[1,5,3,4,2] => [5,3,4,2,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,5,4,2,3] => [4,5,3,2,1] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
[1,5,4,3,2] => [5,4,3,2,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1
[2,1,3,4,5] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,5,4] => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,4,3,5] => [1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,1,4,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,1,5,3,4] => [1,4,5,3,2] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,1,5,4,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,1,5,4] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,1,3,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[2,4,3,1,5] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,3,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,5,1,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,1,4,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[2,5,3,1,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,5,3,4,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,4,3,1] => [1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[3,1,2,4,5] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[3,1,2,5,4] => [3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[3,1,4,2,5] => [4,1,3,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0
[3,1,5,4,2] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[3,2,1,4,5] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,1,5,4] => [2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[3,2,4,1,5] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,4,5,1] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,5,4,1] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,3,1,5] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,3,5,1] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,3,4,1] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,1,3,4,5,6] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,1,4,5,6] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,1,5,6] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,5,1,6] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,5,6,1] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,5,3,4,1,6] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,5,3,4,6,1] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,6,3,4,5,1] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,1,4,5,6] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,4,1,5,6] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,4,5,1,6] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,4,5,6,1] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,4,6,5,1] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,5,4,1,6] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,2,5,4,6,1] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001603
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [-1,-2] => []
=> ? = 0 + 1
[1,2,3] => [1,2,3] => [-1,-2,-3] => []
=> ? = 0 + 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> ? = 0 + 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> ? = 0 + 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => []
=> ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> ? = 0 + 1
[1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => []
=> ? = 0 + 1
[1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => []
=> ? = 0 + 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => []
=> ? = 0 + 1
[2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => []
=> ? = 0 + 1
[3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => []
=> ? = 0 + 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> ? = 0 + 1
[3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => []
=> ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => []
=> ? = 0 + 1
[4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => []
=> ? = 0 + 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => []
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> ? = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => []
=> ? = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => []
=> ? = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> ? = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> ? = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => []
=> ? = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => []
=> ? = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => []
=> ? = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> ? = 0 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => []
=> ? = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => []
=> ? = 0 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => []
=> ? = 0 + 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> ? = 0 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> ? = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> ? = 1 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => []
=> ? = 0 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> ? = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 1 = 0 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 1 = 0 + 1
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 1 = 0 + 1
[2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => []
=> ? = 0 + 1
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 1 = 0 + 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> ? = 1 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 1 = 0 + 1
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 1 = 0 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => []
=> ? = 0 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> ? = 1 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [-3,-1,-2,-4,-5] => []
=> ? = 0 + 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> ? = 1 + 1
[3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => [4]
=> 1 = 0 + 1
[3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => [4]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => [2]
=> ? = 0 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [2,2]
=> 2 = 1 + 1
[3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => []
=> ? = 0 + 1
[3,2,4,5,1] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => [4]
=> 1 = 0 + 1
[3,2,5,1,4] => [3,2,5,1,4] => [-3,-2,-5,-1,-4] => [4]
=> 1 = 0 + 1
[3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => []
=> ? = 0 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [2,2]
=> 2 = 1 + 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [2]
=> ? = 1 + 1
[3,4,2,1,5] => [3,4,2,1,5] => [-3,-4,-2,-1,-5] => [4]
=> 1 = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [2,2]
=> 2 = 1 + 1
[3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => [4]
=> 1 = 0 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [-4,-1,-2,-3,-5] => [4]
=> 1 = 0 + 1
[4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => [4]
=> 1 = 0 + 1
[4,2,1,5,3] => [4,2,1,5,3] => [-4,-2,-1,-5,-3] => [4]
=> 1 = 0 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [2,2]
=> 2 = 1 + 1
[4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => [4]
=> 1 = 0 + 1
[4,3,1,2,5] => [4,3,1,2,5] => [-4,-3,-1,-2,-5] => [4]
=> 1 = 0 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [2,2]
=> 2 = 1 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [2,2]
=> 2 = 1 + 1
[4,5,3,2,1] => [4,5,3,2,1] => [-4,-5,-3,-2,-1] => [4]
=> 1 = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => [-5,-1,-2,-4,-3] => [4]
=> 1 = 0 + 1
[5,1,3,2,4] => [5,1,3,2,4] => [-5,-1,-3,-2,-4] => [4]
=> 1 = 0 + 1
[5,2,1,3,4] => [5,2,1,3,4] => [-5,-2,-1,-3,-4] => [4]
=> 1 = 0 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => [4]
=> 1 = 0 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [2,2]
=> 2 = 1 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [-5,-3,-1,-4,-2] => [4]
=> 1 = 0 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [2,2]
=> 2 = 1 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [-5,-4,-3,-1,-2] => [4]
=> 1 = 0 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [2,2]
=> 2 = 1 + 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [-1,-2] => []
=> ? = 0 + 1
[1,2,3] => [1,2,3] => [-1,-2,-3] => []
=> ? = 0 + 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> ? = 0 + 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> ? = 0 + 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => []
=> ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> ? = 0 + 1
[1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => []
=> ? = 0 + 1
[1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => []
=> ? = 0 + 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => []
=> ? = 0 + 1
[2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => []
=> ? = 0 + 1
[3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => []
=> ? = 0 + 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> ? = 0 + 1
[3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => []
=> ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => []
=> ? = 0 + 1
[4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => []
=> ? = 0 + 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => []
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> ? = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => []
=> ? = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => []
=> ? = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> ? = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> ? = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => []
=> ? = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => []
=> ? = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => []
=> ? = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> ? = 0 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => []
=> ? = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => []
=> ? = 0 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => []
=> ? = 0 + 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> ? = 0 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> ? = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> ? = 1 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => []
=> ? = 0 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> ? = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 1 = 0 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 1 = 0 + 1
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 1 = 0 + 1
[2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => []
=> ? = 0 + 1
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 1 = 0 + 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> ? = 1 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 1 = 0 + 1
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 1 = 0 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => []
=> ? = 0 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> ? = 1 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [-3,-1,-2,-4,-5] => []
=> ? = 0 + 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> ? = 1 + 1
[3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => [4]
=> 1 = 0 + 1
[3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => [4]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => [2]
=> ? = 0 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [2,2]
=> 2 = 1 + 1
[3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => []
=> ? = 0 + 1
[3,2,4,5,1] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => [4]
=> 1 = 0 + 1
[3,2,5,1,4] => [3,2,5,1,4] => [-3,-2,-5,-1,-4] => [4]
=> 1 = 0 + 1
[3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => []
=> ? = 0 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [2,2]
=> 2 = 1 + 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [2]
=> ? = 1 + 1
[3,4,2,1,5] => [3,4,2,1,5] => [-3,-4,-2,-1,-5] => [4]
=> 1 = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [2,2]
=> 2 = 1 + 1
[3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => [4]
=> 1 = 0 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [-4,-1,-2,-3,-5] => [4]
=> 1 = 0 + 1
[4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => [4]
=> 1 = 0 + 1
[4,2,1,5,3] => [4,2,1,5,3] => [-4,-2,-1,-5,-3] => [4]
=> 1 = 0 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [2,2]
=> 2 = 1 + 1
[4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => [4]
=> 1 = 0 + 1
[4,3,1,2,5] => [4,3,1,2,5] => [-4,-3,-1,-2,-5] => [4]
=> 1 = 0 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [2,2]
=> 2 = 1 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [2,2]
=> 2 = 1 + 1
[4,5,3,2,1] => [4,5,3,2,1] => [-4,-5,-3,-2,-1] => [4]
=> 1 = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => [-5,-1,-2,-4,-3] => [4]
=> 1 = 0 + 1
[5,1,3,2,4] => [5,1,3,2,4] => [-5,-1,-3,-2,-4] => [4]
=> 1 = 0 + 1
[5,2,1,3,4] => [5,2,1,3,4] => [-5,-2,-1,-3,-4] => [4]
=> 1 = 0 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => [4]
=> 1 = 0 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [2,2]
=> 2 = 1 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [-5,-3,-1,-4,-2] => [4]
=> 1 = 0 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [2,2]
=> 2 = 1 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [-5,-4,-3,-1,-2] => [4]
=> 1 = 0 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [2,2]
=> 2 = 1 + 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000632
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Values
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 0
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 0
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 0
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 0
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 0
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 0
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 0
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 0
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 0
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 0
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? = 0
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? = 0
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 1
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? = 0
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? = 0
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? = 0
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,5),(1,10),(1,33),(2,9),(2,34),(3,6),(3,7),(3,38),(4,25),(5,3),(5,8),(5,25),(6,19),(6,30),(7,12),(7,19),(7,37),(8,13),(8,35),(8,38),(9,18),(9,31),(10,11),(10,29),(10,36),(11,26),(11,28),(12,22),(12,29),(13,21),(13,32),(14,45),(15,44),(16,46),(17,41),(18,42),(19,2),(19,39),(20,27),(21,20),(22,17),(22,40),(23,17),(23,46),(24,14),(25,1),(25,35),(26,18),(26,43),(27,15),(27,41),(28,15),(28,43),(29,26),(29,40),(30,16),(30,39),(31,14),(31,42),(32,16),(32,23),(33,20),(33,36),(34,24),(34,31),(35,21),(35,33),(36,27),(36,28),(36,40),(37,22),(37,23),(37,39),(38,30),(38,32),(38,37),(39,34),(39,46),(40,41),(40,43),(41,44),(42,45),(43,42),(43,44),(44,45),(46,24)],47)
=> ? = 0
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 1
[3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,11),(0,12),(1,22),(2,19),(3,18),(3,23),(4,14),(4,20),(5,15),(6,15),(6,17),(7,5),(8,1),(8,17),(9,3),(9,21),(9,26),(10,2),(10,16),(11,13),(12,7),(13,6),(13,8),(14,29),(15,24),(16,19),(17,9),(17,22),(17,24),(18,20),(18,28),(20,10),(20,29),(21,23),(21,27),(22,25),(22,26),(23,28),(24,21),(24,25),(25,27),(26,4),(26,18),(26,27),(27,14),(27,28),(28,29),(29,16)],30)
=> ? = 0
[3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? = 0
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,9),(0,19),(1,42),(2,4),(2,20),(2,29),(3,50),(4,5),(4,52),(5,3),(5,53),(6,11),(6,35),(7,12),(7,31),(8,7),(8,34),(9,2),(9,33),(10,18),(10,46),(10,48),(11,27),(11,28),(12,25),(12,47),(13,15),(13,43),(13,44),(14,16),(14,26),(14,45),(15,17),(15,41),(15,51),(16,24),(16,40),(17,23),(17,39),(18,22),(18,32),(19,33),(19,34),(20,38),(20,47),(20,52),(21,55),(22,54),(23,57),(24,56),(25,62),(26,6),(26,63),(27,58),(28,58),(29,1),(29,38),(30,27),(30,60),(31,25),(32,26),(32,54),(33,29),(34,31),(35,28),(36,43),(37,21),(37,61),(38,42),(38,62),(39,30),(39,57),(40,30),(40,56),(41,23),(41,59),(42,13),(42,36),(43,41),(43,55),(44,51),(44,55),(45,24),(45,63),(46,22),(46,61),(47,49),(47,62),(48,14),(48,32),(48,61),(49,37),(49,46),(50,21),(50,44),(51,39),(51,40),(51,59),(52,10),(52,49),(52,53),(53,37),(53,48),(53,50),(54,63),(55,59),(56,60),(57,60),(59,56),(59,57),(60,58),(61,45),(61,54),(62,36),(63,35)],64)
=> ? = 0
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 1
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 0
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 0
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001397
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001397: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001397: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 33%
Values
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 0
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 0
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 0
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 0
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 0
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 0
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 0
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 0
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 0
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 0
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? = 0
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 0
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? = 0
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 1
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? = 0
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? = 0
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? = 0
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,5),(1,10),(1,33),(2,9),(2,34),(3,6),(3,7),(3,38),(4,25),(5,3),(5,8),(5,25),(6,19),(6,30),(7,12),(7,19),(7,37),(8,13),(8,35),(8,38),(9,18),(9,31),(10,11),(10,29),(10,36),(11,26),(11,28),(12,22),(12,29),(13,21),(13,32),(14,45),(15,44),(16,46),(17,41),(18,42),(19,2),(19,39),(20,27),(21,20),(22,17),(22,40),(23,17),(23,46),(24,14),(25,1),(25,35),(26,18),(26,43),(27,15),(27,41),(28,15),(28,43),(29,26),(29,40),(30,16),(30,39),(31,14),(31,42),(32,16),(32,23),(33,20),(33,36),(34,24),(34,31),(35,21),(35,33),(36,27),(36,28),(36,40),(37,22),(37,23),(37,39),(38,30),(38,32),(38,37),(39,34),(39,46),(40,41),(40,43),(41,44),(42,45),(43,42),(43,44),(44,45),(46,24)],47)
=> ? = 0
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 0
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 1
[3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? = 0
[3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,11),(0,12),(1,22),(2,19),(3,18),(3,23),(4,14),(4,20),(5,15),(6,15),(6,17),(7,5),(8,1),(8,17),(9,3),(9,21),(9,26),(10,2),(10,16),(11,13),(12,7),(13,6),(13,8),(14,29),(15,24),(16,19),(17,9),(17,22),(17,24),(18,20),(18,28),(20,10),(20,29),(21,23),(21,27),(22,25),(22,26),(23,28),(24,21),(24,25),(25,27),(26,4),(26,18),(26,27),(27,14),(27,28),(28,29),(29,16)],30)
=> ? = 0
[3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? = 0
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,9),(0,19),(1,42),(2,4),(2,20),(2,29),(3,50),(4,5),(4,52),(5,3),(5,53),(6,11),(6,35),(7,12),(7,31),(8,7),(8,34),(9,2),(9,33),(10,18),(10,46),(10,48),(11,27),(11,28),(12,25),(12,47),(13,15),(13,43),(13,44),(14,16),(14,26),(14,45),(15,17),(15,41),(15,51),(16,24),(16,40),(17,23),(17,39),(18,22),(18,32),(19,33),(19,34),(20,38),(20,47),(20,52),(21,55),(22,54),(23,57),(24,56),(25,62),(26,6),(26,63),(27,58),(28,58),(29,1),(29,38),(30,27),(30,60),(31,25),(32,26),(32,54),(33,29),(34,31),(35,28),(36,43),(37,21),(37,61),(38,42),(38,62),(39,30),(39,57),(40,30),(40,56),(41,23),(41,59),(42,13),(42,36),(43,41),(43,55),(44,51),(44,55),(45,24),(45,63),(46,22),(46,61),(47,49),(47,62),(48,14),(48,32),(48,61),(49,37),(49,46),(50,21),(50,44),(51,39),(51,40),(51,59),(52,10),(52,49),(52,53),(53,37),(53,48),(53,50),(54,63),(55,59),(56,60),(57,60),(59,56),(59,57),(60,58),(61,45),(61,54),(62,36),(63,35)],64)
=> ? = 0
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 1
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 1
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 0
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 0
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.
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