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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,1]
 => [1]
 => 0
[1,-2] => [1,2] => [1,1]
 => [1]
 => 0
[-1,2] => [1,2] => [1,1]
 => [1]
 => 0
[-1,-2] => [1,2] => [1,1]
 => [1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,-3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,-2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,-2,-3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[-1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[-1,2,-3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[-1,-2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[-1,-2,-3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[1,3,-2] => [1,3,2] => [2,1]
 => [1]
 => 0
[1,-3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[1,-3,-2] => [1,3,2] => [2,1]
 => [1]
 => 0
[-1,3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[-1,3,-2] => [1,3,2] => [2,1]
 => [1]
 => 0
[-1,-3,2] => [1,3,2] => [2,1]
 => [1]
 => 0
[-1,-3,-2] => [1,3,2] => [2,1]
 => [1]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,1,-3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,-1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[2,-1,-3] => [2,1,3] => [2,1]
 => [1]
 => 0
[-2,1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[-2,1,-3] => [2,1,3] => [2,1]
 => [1]
 => 0
[-2,-1,3] => [2,1,3] => [2,1]
 => [1]
 => 0
[-2,-1,-3] => [2,1,3] => [2,1]
 => [1]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[3,2,-1] => [3,2,1] => [2,1]
 => [1]
 => 0
[3,-2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[3,-2,-1] => [3,2,1] => [2,1]
 => [1]
 => 0
[-3,2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[-3,2,-1] => [3,2,1] => [2,1]
 => [1]
 => 0
[-3,-2,1] => [3,2,1] => [2,1]
 => [1]
 => 0
[-3,-2,-1] => [3,2,1] => [2,1]
 => [1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000319

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[1,-2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[-1,2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[-1,-2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,-2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,-2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,-2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,-2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,-3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,-3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,-3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,-3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,-1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,-1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,-1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,-1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,-2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,-2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,-2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,-2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 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

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[1,-2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[-1,2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[-1,-2] => [1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,-2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,-2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,-2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[-1,-2,-3] => [1,2,3] => [1,1,1]
 => [3]
 => 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,-3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,-3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,-3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-1,-3,-2] => [1,3,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,-1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[2,-1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,-1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-2,-1,-3] => [2,1,3] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,-2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[3,-2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,-2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[-3,-2,-1] => [3,2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3 = 2 + 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: St000031

Mapping

Mp00161: Signed permutations —reverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%

Values

[1,2] => [2,1] => [2,1] => [1,2] => 2 = 0 + 2
[1,-2] => [-2,1] => [2,1] => [1,2] => 2 = 0 + 2
[-1,2] => [2,-1] => [2,1] => [1,2] => 2 = 0 + 2
[-1,-2] => [-2,-1] => [2,1] => [1,2] => 2 = 0 + 2
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[1,2,-3] => [-3,2,1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[1,-2,3] => [3,-2,1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[1,-2,-3] => [-3,-2,1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[-1,2,3] => [3,2,-1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[-1,2,-3] => [-3,2,-1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[-1,-2,3] => [3,-2,-1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[-1,-2,-3] => [-3,-2,-1] => [3,2,1] => [1,2,3] => 3 = 1 + 2
[1,3,2] => [2,3,1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[1,3,-2] => [-2,3,1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[1,-3,2] => [2,-3,1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[1,-3,-2] => [-2,-3,1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[-1,3,2] => [2,3,-1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[-1,3,-2] => [-2,3,-1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[-1,-3,2] => [2,-3,-1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[-1,-3,-2] => [-2,-3,-1] => [2,3,1] => [2,1,3] => 2 = 0 + 2
[2,1,3] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[2,1,-3] => [-3,1,2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[2,-1,3] => [3,-1,2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[2,-1,-3] => [-3,-1,2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[-2,1,3] => [3,1,-2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[-2,1,-3] => [-3,1,-2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[-2,-1,3] => [3,-1,-2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[-2,-1,-3] => [-3,-1,-2] => [3,1,2] => [1,3,2] => 2 = 0 + 2
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[3,2,-1] => [-1,2,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[3,-2,1] => [1,-2,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[3,-2,-1] => [-1,-2,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[-3,2,1] => [1,2,-3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[-3,2,-1] => [-1,2,-3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[-3,-2,1] => [1,-2,-3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[-3,-2,-1] => [-1,-2,-3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,-4] => [-4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,2,-3,4] => [4,-3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,2,-3,-4] => [-4,-3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,-2,3,4] => [4,3,-2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,-2,3,-4] => [-4,3,-2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,-2,-3,4] => [4,-3,-2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,-2,-3,-4] => [-4,-3,-2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,2,3,4] => [4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,2,3,-4] => [-4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,2,-3,4] => [4,-3,2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,2,-3,-4] => [-4,-3,2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,-2,3,4] => [4,3,-2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[-1,-2,3,-4] => [-4,3,-2,-1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ? => ? => ? = 4 + 2
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ? => ? => ? = 2 + 2
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ? => ? => ? = 2 + 2
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ? => ? => ? = 2 + 2
[1,4,2,3,5,6] => [6,5,3,2,4,1] => ? => ? => ? = 2 + 2
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ? => ? => ? = 0 + 2
[1,4,5,2,3,6] => [6,3,2,5,4,1] => ? => ? => ? = 2 + 2
[1,4,5,6,2,3] => [3,2,6,5,4,1] => ? => ? => ? = 0 + 2
[1,5,2,3,4,6] => [6,4,3,2,5,1] => ? => ? => ? = 1 + 2
[1,5,6,2,3,4] => [4,3,2,6,5,1] => ? => ? => ? = 0 + 2
[1,6,2,3,4,5] => [5,4,3,2,6,1] => ? => ? => ? = 0 + 2
[1,6,2,5,3,4] => [4,3,5,2,6,1] => ? => ? => ? = 0 + 2
[1,6,4,2,3,5] => [5,3,2,4,6,1] => ? => ? => ? = 0 + 2
[1,6,4,5,2,3] => [3,2,5,4,6,1] => ? => ? => ? = 0 + 2
[3,1,2,4,5,6] => [6,5,4,2,1,3] => ? => ? => ? = 2 + 2
[3,1,2,6,4,5] => [5,4,6,2,1,3] => ? => ? => ? = 0 + 2
[3,1,5,2,4,6] => [6,4,2,5,1,3] => ? => ? => ? = 0 + 2
[3,1,5,6,2,4] => [4,2,6,5,1,3] => ? => ? => ? = 0 + 2
[3,4,1,2,5,6] => [6,5,2,1,4,3] => ? => ? => ? = 2 + 2
[3,4,1,6,2,5] => [5,2,6,1,4,3] => ? => ? => ? = 0 + 2
[3,4,5,1,2,6] => [6,2,1,5,4,3] => ? => ? => ? = 0 + 2
[3,4,5,6,1,2] => [2,1,6,5,4,3] => ? => ? => ? = 0 + 2
[3,5,1,2,4,6] => [6,4,2,1,5,3] => ? => ? => ? = 1 + 2
[3,5,6,1,2,4] => [4,2,1,6,5,3] => ? => ? => ? = 0 + 2
[3,6,1,2,4,5] => [5,4,2,1,6,3] => ? => ? => ? = 0 + 2
[3,6,1,5,2,4] => [4,2,5,1,6,3] => ? => ? => ? = 0 + 2
[3,6,4,1,2,5] => [5,2,1,4,6,3] => ? => ? => ? = 0 + 2
[3,6,4,5,1,2] => [2,1,5,4,6,3] => ? => ? => ? = 0 + 2
[4,1,2,3,5,6] => [6,5,3,2,1,4] => ? => ? => ? = 1 + 2
[4,5,1,2,3,6] => [6,3,2,1,5,4] => ? => ? => ? = 0 + 2
[4,5,6,1,2,3] => [3,2,1,6,5,4] => ? => ? => ? = 2 + 2
[4,6,5,1,2,3] => [3,2,1,5,6,4] => ? => ? => ? = 0 + 2
[5,1,2,3,4,6] => [6,4,3,2,1,5] => ? => ? => ? = 0 + 2
[5,1,2,6,3,4] => [4,3,6,2,1,5] => ? => ? => ? = 0 + 2
[5,1,4,2,3,6] => [6,3,2,4,1,5] => ? => ? => ? = 0 + 2
[5,1,4,6,2,3] => [3,2,6,4,1,5] => ? => ? => ? = 0 + 2
[5,3,1,2,4,6] => [6,4,2,1,3,5] => ? => ? => ? = 0 + 2
[5,3,1,6,2,4] => [4,2,6,1,3,5] => ? => ? => ? = 0 + 2
[5,3,4,1,2,6] => [6,2,1,4,3,5] => ? => ? => ? = 0 + 2
[5,3,4,6,1,2] => [2,1,6,4,3,5] => ? => ? => ? = 0 + 2
[5,6,1,2,3,4] => [4,3,2,1,6,5] => ? => ? => ? = 0 + 2
[5,6,1,4,2,3] => [3,2,4,1,6,5] => ? => ? => ? = 0 + 2
[5,6,3,1,2,4] => [4,2,1,3,6,5] => ? => ? => ? = 0 + 2
[5,6,3,4,1,2] => [2,1,4,3,6,5] => ? => ? => ? = 2 + 2
[6,2,1,5,3,4] => [4,3,5,1,2,6] => ? => ? => ? = 0 + 2
[6,2,4,1,3,5] => [5,3,1,4,2,6] => ? => ? => ? = 0 + 2
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? => ? => ? = 3 + 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => ? => ? => ? = 1 + 2
[1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? => ? => ? = 1 + 2
[1,2,7,3,6,4,5] => [5,4,6,3,7,2,1] => ? => ? => ? = 1 + 2

Description

The number of cycles in the cycle decomposition of a permutation.

Matching statistic: St000942

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%

Values

[1,2] => [1,2] => [1,2] => [1,2] => 2 = 0 + 2
[1,-2] => [1,2] => [1,2] => [1,2] => 2 = 0 + 2
[-1,2] => [1,2] => [1,2] => [1,2] => 2 = 0 + 2
[-1,-2] => [1,2] => [1,2] => [1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[-1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[-1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[-1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[-1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,3,-2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,-3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,-3,-2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[-1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[-1,3,-2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[-1,-3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[-1,-3,-2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[2,1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[2,-1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[2,-1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[-2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[-2,1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[-2,-1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[-2,-1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[3,2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[3,-2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[3,-2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[-3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[-3,2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[-3,-2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[-3,-2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,-2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[-1,-2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[-1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[-1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2

Description

The number of critical left to right maxima of the parking functions. An entry

$p$ in a parking function is critical, if there are exactly

$p-1$ entries smaller than

$p$ and

$n-p$ entries larger than

$p$ . It is a left to right maximum, if there are no larger entries before it. This statistic allows the computation of the Tutte polynomial of the complete graph

$K_{n+1}$ , via

$\sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P},$ where the sum is over all parking functions of length

$n$ , see [1, thm.13.5.16].

Matching statistic: St001857
(load all 5 compositions to match this statistic)

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%

Values

[1,2] => [1,2] => [2,1] => [2,1] => 0
[1,-2] => [1,2] => [2,1] => [2,1] => 0
[-1,2] => [1,2] => [2,1] => [2,1] => 0
[-1,-2] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,2,-3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,-2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,-2,-3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[-1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[-1,2,-3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[-1,-2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[-1,-2,-3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,-3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,-3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,-3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,-3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,1,-3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,-1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,-1,-3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[-2,1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[-2,1,-3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[-2,-1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[-2,-1,-3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[3,2,-1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[3,-2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[3,-2,-1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[-3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[-3,2,-1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[-3,-2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[-3,-2,-1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,2,3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,2,-3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,2,-3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,-2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,-2,3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,-2,-3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,-2,-3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,2,3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,2,-3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,2,-3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,-2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,-2,3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,-2,-3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[-1,-2,-3,-4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,2,4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,2,-4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,2,-4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,-2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,-2,4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,-2,-4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,-2,-4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,2,4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,2,-4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,2,-4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,-2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,-2,4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,-2,-4,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[-1,-2,-4,-3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,3,2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,3,-2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,3,-2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,-3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,-3,2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,-3,-2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,-3,-2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,3,2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,3,-2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,3,-2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,-3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,-3,2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,-3,-2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[-1,-3,-2,-4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => ? = 0
[1,3,4,-2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => ? = 0

Description

The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation

$\pi$ has the reduced words of

$\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.