Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001444
(load all 12 compositions to match this statistic)
Mapping
St001444: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => 0
[(1,2),(3,4)]
 => 0
[(1,3),(2,4)]
 => 1
[(1,4),(2,3)]
 => 0
[(1,2),(3,4),(5,6)]
 => 0
[(1,3),(2,4),(5,6)]
 => 1
[(1,4),(2,3),(5,6)]
 => 0
[(1,5),(2,3),(4,6)]
 => 1
[(1,6),(2,3),(4,5)]
 => 0
[(1,6),(2,4),(3,5)]
 => 1
[(1,5),(2,4),(3,6)]
 => 1
[(1,4),(2,5),(3,6)]
 => 1
[(1,3),(2,5),(4,6)]
 => 1
[(1,2),(3,5),(4,6)]
 => 1
[(1,2),(3,6),(4,5)]
 => 0
[(1,3),(2,6),(4,5)]
 => 1
[(1,4),(2,6),(3,5)]
 => 1
[(1,5),(2,6),(3,4)]
 => 1
[(1,6),(2,5),(3,4)]
 => 0
[(1,2),(3,4),(5,6),(7,8)]
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => 0
[(1,5),(2,3),(4,6),(7,8)]
 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => 1
[(1,6),(2,7),(3,4),(5,8)]
 => 1
[(1,5),(2,7),(3,4),(6,8)]
 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => 1
[(1,2),(3,8),(4,5),(6,7)]
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => 1
Description
The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. Two pairs $(a,b)$ and $(c,d)$ (with $a < b$ and $c < d$) in a perfect matching cross if and only if $a < c < b < d$ or $c < a < d < b$. Define a skew symmetric matrix $M$ whose rows and columns are indexed by the pairs of the matching, with $$ M_{(a,b),(c,d)} = \begin{cases} 1 &\text{if \(a < c < b < d\)}\\ -1 &\text{if \(c < a < d < b\)}\\ 0 &\text{otherwise} \end{cases} $$ The rank of this matrix is always even. The present statistic is half of the matrix' rank.
Matching statistic: St001394
(load all 18 compositions to match this statistic)
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => 1
[(1,4),(2,3)]
 => [4,3,2,1] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => ? = 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [5,3,2,6,1,4,8,7,10,9] => ? = 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [7,3,2,5,4,8,1,6,10,9] => ? = 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [9,3,2,5,4,7,6,10,1,8] => ? = 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => [10,4,5,2,3,7,6,9,8,1] => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => [9,4,5,2,3,7,6,10,1,8] => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [8,4,5,2,3,7,6,1,10,9] => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [7,4,5,2,3,8,1,6,10,9] => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [6,4,5,2,3,1,8,7,10,9] => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [5,4,6,2,1,3,8,7,10,9] => ? = 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [4,5,6,1,2,3,8,7,10,9] => ? = 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [3,5,1,6,2,4,8,7,10,9] => ? = 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [2,1,5,6,3,4,8,7,10,9] => ? = 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [3,6,1,5,4,2,8,7,10,9] => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [4,6,5,1,3,2,8,7,10,9] => ? = 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [5,6,4,3,1,2,8,7,10,9] => ? = 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => [7,5,4,3,2,8,1,6,10,9] => ? = 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => [9,5,4,3,2,7,6,10,1,8] => ? = 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => [10,6,4,3,7,2,5,9,8,1] => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => [9,6,4,3,7,2,5,10,1,8] => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => [8,6,4,3,7,2,5,1,10,9] => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => [7,6,4,3,8,2,1,5,10,9] => ? = 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => [6,7,4,3,8,1,2,5,10,9] => ? = 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [5,7,4,3,1,8,2,6,10,9] => ? = 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [4,7,5,1,3,8,2,6,10,9] => ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => [3,7,1,5,4,8,2,6,10,9] => ? = 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => [2,1,7,5,4,8,3,6,10,9] => ? = 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => [3,8,1,5,4,7,6,2,10,9] => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [4,8,5,1,3,7,6,2,10,9] => ? = 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [5,8,4,3,1,7,6,2,10,9] => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => [6,8,4,3,7,1,5,2,10,9] => ? = 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => [7,8,4,3,6,5,1,2,10,9] => ? = 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => [9,7,4,3,6,5,2,10,1,8] => ? = 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => [10,8,4,3,6,5,9,2,7,1] => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => [9,8,4,3,6,5,10,2,1,7] => ? = 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => [8,9,4,3,6,5,10,1,2,7] => ? = 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => [7,9,4,3,6,5,1,10,2,8] => ? = 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => [6,9,4,3,7,1,5,10,2,8] => ? = 2
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => [5,9,4,3,1,7,6,10,2,8] => ? = 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
 => [4,9,5,1,3,7,6,10,2,8] => ? = 2
[(1,3),(2,9),(4,5),(6,7),(8,10)]
 => [3,9,1,5,4,7,6,10,2,8] => ? = 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
 => [2,1,9,5,4,7,6,10,3,8] => ? = 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
 => [3,10,1,5,4,7,6,9,8,2] => ? = 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
 => [4,10,5,1,3,7,6,9,8,2] => ? = 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
 => [5,10,4,3,1,7,6,9,8,2] => ? = 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
 => [6,10,4,3,7,1,5,9,8,2] => ? = 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
 => [7,10,4,3,6,5,1,9,8,2] => ? = 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
 => [8,10,4,3,6,5,9,1,7,2] => ? = 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
 => [9,10,4,3,6,5,8,7,1,2] => ? = 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
 => [10,9,5,6,3,4,8,7,2,1] => ? = 1
Description
The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$ n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ), $$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.