searching the database
Your data matches 39 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000789
St000789: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 1
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 2
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 2
[(1,5),(2,4),(3,6)]
=> 2
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 2
[(1,2),(3,6),(4,5)]
=> 2
[(1,3),(2,6),(4,5)]
=> 2
[(1,4),(2,6),(3,5)]
=> 2
[(1,5),(2,6),(3,4)]
=> 2
[(1,6),(2,5),(3,4)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> 6
[(1,6),(2,3),(4,5),(7,8)]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> 6
[(1,7),(2,4),(3,5),(6,8)]
=> 8
[(1,6),(2,4),(3,5),(7,8)]
=> 4
[(1,5),(2,4),(3,6),(7,8)]
=> 4
[(1,4),(2,5),(3,6),(7,8)]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> 6
[(1,4),(2,6),(3,5),(7,8)]
=> 4
[(1,5),(2,6),(3,4),(7,8)]
=> 4
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 6
[(1,8),(2,5),(3,4),(6,7)]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> 5
[(1,6),(2,7),(3,4),(5,8)]
=> 5
[(1,5),(2,7),(3,4),(6,8)]
=> 8
[(1,4),(2,7),(3,5),(6,8)]
=> 6
[(1,3),(2,7),(4,5),(6,8)]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> 6
[(1,2),(3,8),(4,5),(6,7)]
=> 3
[(1,3),(2,8),(4,5),(6,7)]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> 8
Description
The number of crossing-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children.
[thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$
Matching statistic: St001115
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Values
[(1,2)]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => [3,1,6,5,4,2] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => [3,2,6,1,5,4] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => [3,2,6,1,5,4] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => [3,2,6,5,4,1] => 1 = 2 - 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => [4,2,6,5,3,1] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => [4,2,6,1,5,3] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,6,5,3,2] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => [2,1,6,5,4,3] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => [2,1,6,5,4,3] => 1 = 2 - 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => [3,1,6,5,4,2] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => [4,1,6,5,3,2] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => [4,3,6,1,5,2] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => [4,3,6,2,5,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [3,2,8,1,7,6,5,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => [3,2,8,1,7,6,5,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => [3,2,8,7,6,1,5,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => [3,2,8,7,6,1,5,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => [3,2,8,7,6,5,4,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => [4,2,8,7,6,5,3,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [4,2,8,7,6,1,5,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => [4,2,8,7,6,1,5,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => [4,2,8,1,7,6,5,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => [4,3,8,1,7,6,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [4,3,8,2,7,1,6,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [4,3,8,2,7,1,6,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [4,3,8,2,7,6,5,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [4,3,8,2,7,6,5,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [4,3,8,2,7,1,6,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => [4,3,8,1,7,6,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => [4,3,8,1,7,6,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [4,3,8,1,7,6,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [4,3,8,1,7,6,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [4,3,8,7,6,1,5,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [4,3,8,7,6,2,5,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [5,3,8,7,6,2,4,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [5,3,8,7,6,1,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [5,3,8,1,7,6,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => [5,1,8,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => [5,1,8,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [5,3,8,1,7,6,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [5,3,8,2,7,1,6,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [5,3,8,2,7,6,4,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [5,2,8,7,6,4,3,1] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [5,2,8,7,6,1,4,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => [5,2,8,1,7,6,4,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [5,1,8,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,1,6,2,7,3,8,5] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,1,6,2,7,4,8,5] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,6,3,7,4,8,5] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => [4,1,8,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,8,6] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,8,5] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,8,4] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,6),(4,8),(5,7)]
=> [2,1,6,8,7,3,5,4] => [2,1,6,3,7,5,8,4] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,7),(4,8),(5,6)]
=> [2,1,7,8,6,5,3,4] => [2,1,6,5,7,3,8,4] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3] => 2 = 3 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => [2,1,5,3,6,4,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,5,4,7,6,9,8,10,3] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [2,1,6,7,8,3,4,5,10,9] => [2,1,6,3,7,4,8,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [2,1,5,7,3,8,4,6,10,9] => [2,1,5,3,7,4,8,6,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [2,1,4,3,7,8,5,6,10,9] => [2,1,4,3,7,5,8,6,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,7,6,8,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,6,5,7,4,8,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,6,5,7,4,9,8,10,3] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,7,6,9,8,10,5] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,6,5,8,7,9,4,10,3] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> [2,1,7,8,9,10,3,4,5,6] => [2,1,7,3,8,4,9,5,10,6] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> [2,1,6,8,9,3,10,4,5,7] => [2,1,6,3,8,4,9,5,10,7] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> [2,1,5,8,3,9,10,4,6,7] => [2,1,5,3,8,4,9,6,10,7] => [2,1,10,9,8,7,6,5,4,3] => 3 = 4 - 1
Description
The number of even descents of a permutation.
Matching statistic: St000891
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[(1,2)]
=> [2,1] => [2,1] => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,4,1,3] => 2 = 1 + 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,4,1,3,6,5] => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,6,1,5,3,2] => 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [5,4,1,6,3,2] => 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [3,1,5,2,6,4] => 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [3,6,2,1,5,4] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,3,6,1,4,5] => 3 = 2 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,6,2,5,1,3] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,4,6,3,5] => 3 = 2 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [5,4,2,6,1,3] => 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [3,2,6,5,1,4] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [2,6,4,3,1,5] => 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,4,1,3,6,5,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [4,6,1,5,3,2,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [6,8,1,5,4,7,3,2] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [7,6,1,3,5,2,8,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [6,8,1,3,5,2,7,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [3,1,5,2,6,4,8,7] => 2 = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [3,6,2,1,5,4,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,3,6,1,4,5,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,6,2,5,1,3,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,4,6,3,5,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [5,4,2,6,1,3,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [3,2,6,5,1,4,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [2,6,4,3,1,5,8,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 3 = 2 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [6,8,1,7,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [5,1,7,2,8,6,4,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [5,8,2,1,7,6,4,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [2,5,8,1,7,4,3,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,8,2,7,4,3,1,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [6,8,2,3,7,5,1,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [6,8,2,5,4,7,1,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,6,8,3,7,5,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [7,6,2,5,4,8,1,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [7,6,2,3,8,5,1,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [7,6,2,8,4,3,1,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [5,2,7,1,8,4,3,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [2,6,5,1,8,4,3,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [4,2,1,6,3,8,7,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [2,4,1,6,3,8,5,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [4,2,7,3,1,8,5,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [4,2,3,7,1,5,8,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [5,2,7,3,8,6,1,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [5,2,7,4,6,8,1,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [2,1,5,3,7,4,8,6] => 2 = 1 + 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [2,1,5,8,4,3,7,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [5,8,4,2,6,7,1,3] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [5,8,3,2,7,6,1,4] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [3,2,4,8,1,5,7,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [2,4,8,3,1,7,5,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [4,8,3,2,1,7,6,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [4,1,7,3,2,8,6,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [4,1,3,7,2,6,8,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [3,1,4,8,2,6,7,5] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [3,4,8,2,1,6,5,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [2,3,4,8,1,5,6,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [3,5,8,2,7,1,4,6] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [4,5,8,2,6,1,3,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,4,5,8,3,6,7] => ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} + 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 3 = 2 + 1
[(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 1
[(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 3 = 2 + 1
[(1,4),(2,8),(3,7),(5,6)]
=> [4,8,7,1,6,5,3,2] => [6,5,3,2,8,7,1,4] => 3 = 2 + 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 3 = 2 + 1
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3 = 2 + 1
[(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3 = 2 + 1
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 3 = 2 + 1
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 3 = 2 + 1
[(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 3 = 2 + 1
[(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 3 = 2 + 1
[(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 3 = 2 + 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 2 = 1 + 1
Description
The number of distinct diagonal sums of a permutation matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
Matching statistic: St000366
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Values
[(1,2)]
=> [2,1] => [1,2] => [1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => [2,1,3,4] => [2,1,3] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [5,6,3,4,1,2] => [5,3,4,1,2] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,3,6,5,1,2] => [4,3,5,1,2] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,5,6,1,2] => [4,3,5,1,2] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [4,2,5,1,6,3] => [4,2,5,1,3] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,2,5,1,3,6] => [4,2,5,1,3] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1,5,4,6] => [3,2,1,5,4] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1,5,6,4] => [3,2,1,5,4] => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,5,4] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,2,6,1,5,3] => [4,2,1,5,3] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [5,6,2,1,4,3] => [5,2,1,4,3] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [5,6,2,1,3,4] => [5,2,1,3,4] => 1 = 2 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [4,2,6,1,3,5] => [4,2,1,3,5] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1,6,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1,4,6,5] => [3,2,1,4,5] => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,2,1,4,5,6] => [3,2,1,4,5] => 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 0 = 1 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [6,5,8,7,3,4,1,2] => [6,5,7,3,4,1,2] => 0 = 1 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => 0 = 1 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [6,4,7,3,8,5,1,2] => [6,4,7,3,5,1,2] => 0 = 1 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,4,7,3,5,8,1,2] => [6,4,7,3,5,1,2] => 0 = 1 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [6,4,7,2,5,1,8,3] => [6,4,7,2,5,1,3] => 0 = 1 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,4,7,2,5,1,3,8] => [6,4,7,2,5,1,3] => 0 = 1 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [5,4,2,7,6,1,3,8] => [5,4,2,7,6,1,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [5,4,2,7,6,1,8,3] => [5,4,2,7,6,1,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [5,4,3,7,6,8,1,2] => [5,4,3,7,6,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [5,4,3,7,8,6,1,2] => [5,4,3,7,6,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,4,3,8,7,6,1,2] => [5,4,3,7,6,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [6,4,8,3,7,5,1,2] => [6,4,3,7,5,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => [7,4,3,6,5,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [6,4,8,3,5,7,1,2] => [6,4,3,5,7,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [5,4,3,8,6,7,1,2] => [5,4,3,6,7,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [5,4,3,6,8,7,1,2] => [5,4,3,6,7,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [5,4,2,6,7,1,8,3] => [5,4,2,6,7,1,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [5,4,2,6,7,1,3,8] => [5,4,2,6,7,1,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [5,3,2,6,1,7,4,8] => [5,3,2,6,1,7,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [5,3,2,6,1,7,8,4] => [5,3,2,6,1,7,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [5,3,2,6,1,8,7,4] => [5,3,2,6,1,7,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [5,4,2,6,8,1,7,3] => [5,4,2,6,1,7,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [5,4,2,8,6,1,7,3] => [5,4,2,6,1,7,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [6,4,8,2,5,1,7,3] => [6,4,2,5,1,7,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [7,8,4,2,5,1,6,3] => [7,4,2,5,1,6,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [7,8,4,2,5,1,3,6] => [7,4,2,5,1,3,6] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [6,4,8,2,5,1,3,7] => [6,4,2,5,1,3,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [5,4,2,8,6,1,3,7] => [5,4,2,6,1,3,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [5,4,2,6,8,1,3,7] => [5,4,2,6,1,3,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [5,3,2,6,1,8,4,7] => [5,3,2,6,1,4,7] => 1 = 2 - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [5,3,2,6,1,4,8,7] => [5,3,2,6,1,4,7] => 1 = 2 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [5,3,2,6,1,4,7,8] => [5,3,2,6,1,4,7] => 1 = 2 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [4,3,2,1,6,5,7,8] => [4,3,2,1,6,5,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [4,3,2,1,6,8,5,7] => [4,3,2,1,6,5,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [4,3,2,1,8,6,5,7] => [4,3,2,1,6,5,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [5,3,2,8,1,6,4,7] => [5,3,2,1,6,4,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [6,3,8,2,1,5,4,7] => [6,3,2,1,5,4,7] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [7,8,3,2,1,5,4,6] => [7,3,2,1,5,4,6] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [7,8,3,2,1,5,6,4] => [7,3,2,1,5,6,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [6,3,8,2,1,5,7,4] => [6,3,2,1,5,7,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [5,3,2,8,1,6,7,4] => [5,3,2,1,6,7,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [4,3,2,1,8,6,7,5] => [4,3,2,1,6,7,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [4,3,2,1,6,8,7,5] => [4,3,2,1,6,7,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [4,3,2,1,6,7,8,5] => [4,3,2,1,6,7,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [4,3,2,1,6,7,5,8] => [4,3,2,1,6,7,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [4,3,2,1,7,6,5,8] => [4,3,2,1,7,6,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [4,3,2,1,7,6,8,5] => [4,3,2,1,7,6,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [4,3,2,1,7,8,6,5] => [4,3,2,1,7,6,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [5,3,2,8,1,7,6,4] => [5,3,2,1,7,6,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [6,3,8,2,1,7,5,4] => [6,3,2,1,7,5,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [7,8,3,2,1,6,5,4] => [7,3,2,1,6,5,4] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [7,8,4,2,6,1,5,3] => [7,4,2,6,1,5,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [6,4,8,2,7,1,5,3] => [6,4,2,7,1,5,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [5,4,2,8,7,1,6,3] => [5,4,2,7,1,6,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [5,4,2,7,8,1,6,3] => [5,4,2,7,1,6,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [7,8,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => 2 = 3 - 1
[(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => [4,3,2,1,8,5,6,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => [4,3,2,1,5,8,6,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => [9,7,8,5,6,3,4,1,2] => 0 = 1 - 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [8,6,9,4,7,2,5,1,3,10] => [8,6,9,4,7,2,5,1,3] => 0 = 1 - 1
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [7,5,3,8,2,6,1,4,9,10] => [7,5,3,8,2,6,1,4,9] => 1 = 2 - 1
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [6,4,3,2,7,1,5,8,9,10] => [6,4,3,2,7,1,5,8,9] => 2 = 3 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => 3 = 4 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 23%
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 23%
Values
[(1,2)]
=> {{1,2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0 = 1 - 1
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 0 = 1 - 1
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,3,5},{2},{4},{6}}
=> {{1,5},{2,3},{4},{6}}
=> 0 = 1 - 1
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> {{1,3,5},{2},{4,6}}
=> {{1,5},{2,3,6},{4}}
=> 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3},{2},{4,6},{5}}
=> {{1,6},{2,3},{4},{5}}
=> 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> {{1,3},{2,4,6},{5}}
=> {{1,4},{2,3,6},{5}}
=> 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1},{2,4,6},{3},{5}}
=> {{1,6},{2,4},{3},{5}}
=> 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> {{1},{2,4,6},{3,5}}
=> {{1,5},{2,4,6},{3}}
=> 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> {{1,4},{2,6},{3,5}}
=> {{1,5},{2,6},{3,4}}
=> 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> {{1,3},{2,5},{4,6}}
=> {{1},{2,3},{4,5,6}}
=> 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> {{1,3,5},{2,4},{6}}
=> {{1,4},{2,3,5},{6}}
=> 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,5},{2,4},{3},{6}}
=> {{1},{2,4},{3,5},{6}}
=> 1 = 2 - 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> {{1,5},{2,4,6},{3}}
=> {{1},{2,4},{3,5,6}}
=> 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> {{1,5},{2,4},{3,6}}
=> {{1,6},{2,4},{3,5}}
=> 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> {{1,3,5},{2,6},{4}}
=> {{1,5},{2,3},{4,6}}
=> 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1},{2,6},{3,5},{4}}
=> {{1},{2,5},{3,6},{4}}
=> 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> {{1,5},{2,3,7},{4},{6},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,4),(5,6),(7,8)]
=> {{1,3},{2,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> {{1,5,8},{2,3,7},{4},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,5},{2},{4},{6,8},{7}}
=> {{1,5},{2,3,8},{4},{6},{7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,3),(4,6),(7,8)]
=> {{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,5},{2},{4,6,8},{7}}
=> {{1,5,8},{2,3,6},{4},{7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3},{2},{4,6,8},{5},{7}}
=> {{1,6},{2,3,8},{4},{5},{7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> {{1,7},{2,3},{4,5},{6,8}}
=> {{1,3},{2,4,6,8},{5},{7}}
=> {{1,4,8},{2,3,6},{5},{7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5},{7}}
=> {{1,6},{2,4,8},{3},{5},{7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> {{1,8},{2,4},{3,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5,7}}
=> {{1,6,8},{2,4,7},{3},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> {{1,7},{2,4},{3,5},{6,8}}
=> {{1,3},{2,8},{4,6},{5,7}}
=> {{1,6},{2,3,7},{4},{5,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> {{1,6},{2,4},{3,5},{7,8}}
=> {{1,3},{2},{4,6,8},{5,7}}
=> {{1,6,8},{2,3,7},{4},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> {{1,5},{2,4},{3,6},{7,8}}
=> {{1,3,6},{2},{4,8},{5,7}}
=> {{1,8},{2,3},{4,6,7},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> {{1,4},{2,5},{3,6},{7,8}}
=> {{1,3,6},{2},{4,7},{5,8}}
=> {{1,7},{2,3},{4,6,8},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,6),(7,8)]
=> {{1,3},{2,5},{4,6},{7,8}}
=> {{1,3,5},{2},{4,7},{6,8}}
=> {{1,5},{2,3,8},{4,7},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> {{1,2},{3,5},{4,6},{7,8}}
=> {{1,3,5,7},{2},{4,6},{8}}
=> {{1,5,7},{2,3,6},{4},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6},{5},{8}}
=> {{1,6},{2,3},{4,7},{5},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,5),(7,8)]
=> {{1,3},{2,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6,8},{5}}
=> {{1,6},{2,3,8},{4,7},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> {{1,4},{2,6},{3,5},{7,8}}
=> {{1,3,7},{2},{4,6},{5,8}}
=> {{1,6},{2,3},{4,7,8},{5}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> {{1,5},{2,6},{3,4},{7,8}}
=> {{1,3,5,7},{2},{4,8},{6}}
=> {{1,5},{2,3,7},{4,8},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,3},{2},{4,8},{5,7},{6}}
=> {{1,7},{2,3},{4,8},{5},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> {{1,7},{2,5},{3,4},{6,8}}
=> {{1,3},{2,4,8},{5,7},{6}}
=> {{1,4},{2,3,7},{5,8},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> {{1},{2,4,8},{3},{5,7},{6}}
=> {{1,7},{2,4},{3,8},{5},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> {{1,8},{2,6},{3,4},{5,7}}
=> {{1},{2,4,8},{3,5,7},{6}}
=> {{1,5},{2,4,7},{3,8},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> {{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2,8},{3,5,7},{6}}
=> {{1,5},{2},{3,4,7},{6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> {{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2,5,7},{3,8},{6}}
=> {{1},{2,5},{3,4,7},{6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> {{1,5},{2,7},{3,4},{6,8}}
=> {{1,3},{2,5,7},{4,8},{6}}
=> {{1,8},{2,3},{4,5,7},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> {{1,4},{2,7},{3,5},{6,8}}
=> {{1,3},{2,7},{4,6},{5,8}}
=> {{1,6},{2,3},{4},{5,7,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> {{1,3},{2,7},{4,5},{6,8}}
=> {{1,3},{2,7},{4,6,8},{5}}
=> {{1,6},{2,3,8},{4},{5,7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> {{1,2},{3,7},{4,5},{6,8}}
=> {{1,3,7},{2,4,6},{5},{8}}
=> {{1,4},{2,3,6},{5,7},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,7},{2,4,6},{3},{5},{8}}
=> {{1,6},{2,4},{3},{5,7},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,5),(6,7)]
=> {{1,3},{2,8},{4,5},{6,7}}
=> {{1,7},{2,4,6,8},{3},{5}}
=> {{1,6},{2,4,8},{3},{5,7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> {{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,4,6},{3},{5,8}}
=> {{1,6},{2,4},{3},{5,7,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> {{1,5},{2,8},{3,4},{6,7}}
=> {{1,5,7},{2,4,8},{3},{6}}
=> {{1,8},{2,4},{3,5,7},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> {{1,6},{2,8},{3,4},{5,7}}
=> {{1,5,7},{2,4},{3,8},{6}}
=> {{1},{2,4},{3,5,7},{6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> {{1,7},{2,8},{3,4},{5,6}}
=> {{1,3,5,7},{2,8},{4},{6}}
=> {{1,5},{2,3,7},{4},{6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> {{1},{2,8},{3,5,7},{4},{6}}
=> {{1,7},{2,5},{3},{4,8},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,8},{3,5,7},{4,6}}
=> {{1,6},{2,5,7},{3,8},{4}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> {{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> {{1,6},{2,5},{3,8},{4,7}}
=> 4 = 5 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> {{1,6},{2,8},{3,5},{4,7}}
=> {{1,7},{2,5},{3,8},{4,6}}
=> {{1,6},{2,8},{3,5},{4,7}}
=> 3 = 4 - 1
[(1,5),(2,8),(3,6),(4,7)]
=> {{1,5},{2,8},{3,6},{4,7}}
=> {{1,7},{2,5},{3,6},{4,8}}
=> {{1,8},{2,6},{3,5},{4,7}}
=> 2 = 3 - 1
[(1,4),(2,8),(3,6),(5,7)]
=> {{1,4},{2,8},{3,6},{5,7}}
=> {{1,7},{2,4},{3,6},{5,8}}
=> {{1},{2,4},{3,6},{5,7,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> {{1,3},{2,8},{4,6},{5,7}}
=> {{1,7},{2,4},{3,5},{6,8}}
=> {{1,5},{2,4,8},{3},{6,7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> {{1,2},{3,8},{4,6},{5,7}}
=> {{1,7},{2,4,6},{3,5},{8}}
=> {{1,5},{2,4,6},{3,7},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,6),(5,8)]
=> {{1,2},{3,7},{4,6},{5,8}}
=> {{1,4,7},{2,6},{3,5},{8}}
=> {{1,5},{2,6},{3,4,7},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,6),(5,8)]
=> {{1,3},{2,7},{4,6},{5,8}}
=> {{1,4},{2,7},{3,5},{6,8}}
=> {{1,5},{2},{3,4,8},{6,7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> {{1,4},{2,7},{3,6},{5,8}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2,6},{3,4},{5,7,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> {{1,5},{2,7},{3,6},{4,8}}
=> {{1,5},{2,7},{3,6},{4,8}}
=> {{1,8},{2,6},{3,7},{4,5}}
=> 1 = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> {{1,6},{2,7},{3,5},{4,8}}
=> {{1,5},{2,7},{3,8},{4,6}}
=> {{1,6},{2,8},{3,7},{4,5}}
=> 2 = 3 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> {{1,7},{2,6},{3,5},{4,8}}
=> {{1,5},{2,8},{3,7},{4,6}}
=> {{1,6},{2,7},{3,8},{4,5}}
=> 3 = 4 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> {{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5,8},{3,7},{4,6}}
=> {{1,6},{2,7},{3,5,8},{4}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> {{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2,5,8},{3,6},{4,7}}
=> {{1,7},{2,6},{3,5,8},{4}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> {{1,7},{2,5},{3,6},{4,8}}
=> {{1,5},{2,8},{3,6},{4,7}}
=> {{1,7},{2,6},{3,8},{4,5}}
=> 2 = 3 - 1
[(1,6),(2,5),(3,7),(4,8)]
=> {{1,6},{2,5},{3,7},{4,8}}
=> {{1,5},{2,6},{3,8},{4,7}}
=> {{1,7},{2,8},{3,6},{4,5}}
=> 1 = 2 - 1
[(1,5),(2,6),(3,7),(4,8)]
=> {{1,5},{2,6},{3,7},{4,8}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> {{1,4},{2,6},{3,7},{5,8}}
=> {{1,4},{2,6},{3,7},{5,8}}
=> {{1},{2,7},{3,4},{5,6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> {{1,3},{2,6},{4,7},{5,8}}
=> {{1,4},{2,5},{3,7},{6,8}}
=> {{1},{2,5},{3,4,8},{6,7}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> {{1,2},{3,6},{4,7},{5,8}}
=> {{1,4,7},{2,5},{3,6},{8}}
=> {{1,6},{2,5},{3,4,7},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,7),(6,8)]
=> {{1,2},{3,5},{4,7},{6,8}}
=> {{1,3,7},{2,5},{4,6},{8}}
=> {{1,7},{2,3},{4,5,6},{8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,7),(6,8)]
=> {{1,3},{2,5},{4,7},{6,8}}
=> {{1,3},{2,5},{4,7},{6,8}}
=> {{1,7},{2,3,8},{4,5},{6}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> {{1,4},{2,5},{3,7},{6,8}}
=> {{1,3},{2,6},{4,7},{5,8}}
=> {{1,7},{2,3},{4},{5,6,8}}
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,8),(4,7)]
=> {{1,5},{2,6},{3,8},{4,7}}
=> {{1,6},{2,5},{3,7},{4,8}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> 1 = 2 - 1
[(1,6),(2,5),(3,8),(4,7)]
=> {{1,6},{2,5},{3,8},{4,7}}
=> {{1,6},{2,5},{3,8},{4,7}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> 2 = 3 - 1
[(1,7),(2,5),(3,8),(4,6)]
=> {{1,7},{2,5},{3,8},{4,6}}
=> {{1,6},{2,8},{3,5},{4,7}}
=> {{1,7},{2,5},{3,8},{4,6}}
=> 3 = 4 - 1
[(1,7),(2,6),(3,8),(4,5)]
=> {{1,7},{2,6},{3,8},{4,5}}
=> {{1,4,6},{2,8},{3,7},{5}}
=> {{1,6},{2,7},{3,4},{5,8}}
=> 3 = 4 - 1
[(1,6),(2,7),(3,8),(4,5)]
=> {{1,6},{2,7},{3,8},{4,5}}
=> {{1,4,6},{2,7},{3,8},{5}}
=> {{1,6},{2,8},{3,4},{5,7}}
=> 2 = 3 - 1
[(1,5),(2,7),(3,8),(4,6)]
=> {{1,5},{2,7},{3,8},{4,6}}
=> {{1,6},{2,7},{3,5},{4,8}}
=> {{1,8},{2,5},{3,7},{4,6}}
=> 2 = 3 - 1
[(1,5),(2,8),(3,7),(4,6)]
=> {{1,5},{2,8},{3,7},{4,6}}
=> {{1,7},{2,6},{3,5},{4,8}}
=> {{1,8},{2,5},{3,6},{4,7}}
=> 3 = 4 - 1
[(1,6),(2,8),(3,7),(4,5)]
=> {{1,6},{2,8},{3,7},{4,5}}
=> {{1,7},{2,4,6},{3,8},{5}}
=> {{1,6},{2,4},{3,8},{5,7}}
=> 3 = 4 - 1
[(1,7),(2,8),(3,6),(4,5)]
=> {{1,7},{2,8},{3,6},{4,5}}
=> {{1,3,7},{2,8},{4,6},{5}}
=> {{1,6},{2,3},{4,7},{5,8}}
=> 3 = 4 - 1
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> 0 = 1 - 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000648
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[(1,2)]
=> [2,1] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [4,3,1,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,3,1,2,6,5] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,2,1,6,5] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,3,2,6,1,4] => 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [5,3,2,6,4,1] => 1 = 2 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [5,4,6,2,3,1] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [5,4,6,2,1,3] => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,4,6,1,2,3] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,3,1,6,2,4] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,3,4] => 0 = 1 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [5,3,1,6,4,2] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [5,4,6,1,3,2] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [5,4,6,3,1,2] => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [5,4,6,3,2,1] => 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,3,1,2,6,5,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 0 = 1 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [5,3,2,6,1,4,8,7] => 1 = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [5,3,2,6,4,1,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [5,3,2,7,4,8,1,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [5,3,2,7,4,8,6,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [5,4,7,2,3,8,6,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [5,4,7,2,3,8,1,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [5,4,6,2,3,1,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [5,4,6,2,1,3,8,7] => 1 = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,4,6,1,2,3,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [5,3,1,6,2,4,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,6,5,3,4,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 0 = 1 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [5,3,1,6,4,2,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [5,4,6,1,3,2,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [5,4,6,3,1,2,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [5,4,6,3,2,1,8,7] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [5,4,7,3,2,8,1,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [5,4,7,3,2,8,6,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [6,4,7,3,8,2,5,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [6,4,7,3,8,2,1,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [6,4,7,3,8,1,2,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [5,4,7,3,1,8,2,6] => 2 = 3 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [5,4,7,1,3,8,2,6] => 2 = 3 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [5,3,1,7,4,8,2,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,7,5,4,8,3,6] => 1 = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,7,5,4,8,6,3] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [5,3,1,7,4,8,6,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [5,4,7,1,3,8,6,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [5,4,7,3,1,8,6,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [6,4,7,3,8,1,5,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [6,4,7,3,8,5,1,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [6,4,7,3,8,5,2,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [6,5,7,8,3,4,2,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [6,5,7,8,3,4,1,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [6,5,7,8,3,1,4,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [6,5,7,8,1,3,4,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [6,4,7,1,8,3,5,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [6,3,1,7,8,4,5,2] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,7,6,8,4,5,3] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,7,6,8,4,3,5] => 1 = 2 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [6,3,1,7,8,4,2,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [6,4,7,1,8,3,2,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [6,5,7,8,1,3,2,4] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [6,5,7,8,3,1,2,4] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [6,5,7,8,3,2,1,4] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [6,5,7,8,3,2,4,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [6,5,7,8,2,3,4,1] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [6,5,7,8,2,3,1,4] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => 0 = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [6,5,7,8,1,2,3,4] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [6,4,7,1,8,2,3,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [6,3,1,7,8,2,4,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,7,6,8,3,4,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,7,5,3,8,4,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [5,3,1,7,2,8,4,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [5,4,7,1,2,8,3,6] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [5,4,7,2,1,8,3,6] => 2 = 3 - 1
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [6,4,7,2,8,1,3,5] => 1 = 2 - 1
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [6,4,7,2,8,3,1,5] => ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,7),(5,8)]
=> [3,6,2,7,8,1,4,5] => [6,3,2,7,8,1,4,5] => 0 = 1 - 1
[(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => [5,3,2,7,1,8,4,6] => 1 = 2 - 1
[(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [4,3,2,1,7,8,5,6] => 2 = 3 - 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 0 = 1 - 1
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000834
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 9%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 9%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,2] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 2
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 2
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 2
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 2
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => 2
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => 2
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [5,1,2,3,4,6,7,8] => 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [1,2,3,5,4,6,7,8] => [5,1,2,3,4,6,7,8] => 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [1,4,3,6,2,5,7,8] => [4,6,3,1,2,5,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => [4,6,3,1,2,5,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [1,4,3,7,2,5,6,8] => [7,4,3,1,2,5,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => [7,4,6,8,3,1,2,5] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [1,4,3,7,5,8,2,6] => [7,4,8,3,5,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [1,4,3,7,2,5,6,8] => [7,4,3,1,2,5,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [1,4,3,7,6,8,2,5] => [7,4,6,8,3,1,2,5] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [1,4,3,7,5,8,2,6] => [7,4,8,3,5,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 2
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 2
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 2
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 2
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8}
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [1,2,3,4,5,7,6,8] => [7,1,2,3,4,5,6,8] => 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [1,2,3,4,5,7,6,8] => [7,1,2,3,4,5,6,8] => 1
[(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 2
[(1,2),(3,6),(4,8),(5,7)]
=> [2,1,6,7,8,3,5,4] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 2
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 1
[(1,2),(3,4),(5,6),(7,9),(8,10)]
=> [2,1,4,3,6,5,9,10,7,8] => [1,2,3,4,5,6,7,9,8,10] => [9,1,2,3,4,5,6,7,8,10] => 1
[(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [1,2,3,4,5,6,7,9,8,10] => [9,1,2,3,4,5,6,7,8,10] => 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St001083
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 9%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 9%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 1 = 2 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 0 = 1 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [5,1,2,3,4,6,7,8] => 0 = 1 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [1,2,3,5,4,6,7,8] => [5,1,2,3,4,6,7,8] => 0 = 1 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [1,3,2,5,4,6,7,8] => [3,5,1,2,4,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [1,4,3,6,2,5,7,8] => [4,6,3,1,2,5,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => [4,6,3,1,2,5,7,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [1,4,3,7,2,5,6,8] => [7,4,3,1,2,5,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => [7,4,6,8,3,1,2,5] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [1,4,3,7,5,8,2,6] => [7,4,8,3,5,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [1,4,3,7,2,5,6,8] => [7,4,3,1,2,5,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 1 = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 1 = 2 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [1,4,3,7,6,8,2,5] => [7,4,6,8,3,1,2,5] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [1,4,3,7,5,8,2,6] => [7,4,8,3,5,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 1 = 2 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 1 = 2 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [6,3,7,1,2,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 1 = 2 - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 1 = 2 - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,5,7,1,2,4,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [1,2,3,4,5,7,6,8] => [7,1,2,3,4,5,6,8] => 0 = 1 - 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [1,2,3,4,5,7,6,8] => [7,1,2,3,4,5,6,8] => 0 = 1 - 1
[(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [1,2,3,5,4,7,6,8] => [5,7,1,2,3,4,6,8] => 1 = 2 - 1
[(1,2),(3,6),(4,8),(5,7)]
=> [2,1,6,7,8,3,5,4] => [1,2,3,6,4,7,5,8] => [1,6,7,2,3,4,5,8] => 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,9),(8,10)]
=> [2,1,4,3,6,5,9,10,7,8] => [1,2,3,4,5,6,7,9,8,10] => [9,1,2,3,4,5,6,7,8,10] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [1,2,3,4,5,6,7,9,8,10] => [9,1,2,3,4,5,6,7,8,10] => 0 = 1 - 1
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000871
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00252: Permutations —restriction⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[(1,2)]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,5] => [3,2,4,1,5] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,4,1,2,5] => [1,4,2,3,5] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [5,3,2,1,4] => [4,3,1,5,2] => 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,1] => [5,3,2,4,1] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,5,2,3,1] => [5,1,4,2,3] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [5,4,2,1,3] => [4,1,5,3,2] => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => [1,2,5,3,4] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,5,1,2,4] => [1,2,4,5,3] => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,4] => [3,2,1,5,4] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,3] => [4,3,5,2,1] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,2] => [4,5,2,3,1] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,5,1,3,2] => [2,5,4,1,3] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7] => [4,3,5,2,6,1,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,4,1,2,6,5,7] => [2,5,3,4,6,1,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => [5,4,3,2,6,1,7] => 2 = 3 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [5,3,2,6,1,4,7] => [1,5,4,6,2,3,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [6,3,2,5,4,1,7] => [6,4,2,5,3,1,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [7,3,2,5,4,1,6] => [6,4,2,5,1,7,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1] => [7,4,3,5,2,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,5,2,3,7,6,1] => [7,2,5,3,4,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [7,4,5,2,3,1,6] => [6,1,5,2,3,7,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [6,4,5,2,3,1,7] => [6,1,5,2,4,3,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [5,4,6,2,1,3,7] => [5,1,6,3,2,4,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,7] => [1,2,6,3,4,5,7] => 1 = 2 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,5,1,6,2,4,7] => [4,1,3,6,2,5,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,6,3,4,7] => [3,2,1,6,4,5,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,7] => [5,4,6,3,2,1,7] => 1 = 2 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,6,1,5,4,2,7] => [3,6,4,5,2,1,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,6,5,1,3,2,7] => [2,6,5,3,4,1,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [5,6,4,3,1,2,7] => [1,6,5,4,2,3,7] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 1 = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [5,4,3,2,7,6,1] => [7,5,4,3,2,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [6,4,3,7,2,5,1] => [7,1,5,4,6,2,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [6,7,4,3,1,2,5] => [1,6,5,2,7,3,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [5,7,4,3,1,2,6] => [1,6,5,2,4,7,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,7,5,1,3,2,6] => [2,6,5,3,1,7,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,7,1,5,4,2,6] => [3,6,4,5,1,7,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,7,5,4,3,6] => [5,4,6,3,1,7,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,3] => [5,4,7,3,2,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,2] => [5,7,3,4,2,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,5,1,3,7,6,2] => [3,7,5,2,4,6,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [5,4,3,1,7,6,2] => [6,7,4,3,2,5,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [6,4,3,7,1,5,2] => [2,7,5,4,6,1,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [7,4,3,6,5,1,2] => [1,7,5,3,6,4,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => [7,6,4,2,5,3,1] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [7,6,1,5,2,4,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [6,5,7,3,1,4,2] => [6,7,1,5,3,2,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [5,6,7,1,3,4,2] => [2,7,1,6,3,4,5] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [4,6,1,7,3,5,2] => [5,7,1,3,6,2,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [3,1,6,7,4,5,2] => [4,7,2,1,6,3,5] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [2,1,6,7,4,5,3] => [4,3,7,1,6,2,5] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [2,1,7,6,4,3,5] => [5,4,6,1,7,3,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [3,7,1,6,4,2,5] => [3,6,4,1,7,5,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [4,7,6,1,3,2,5] => [2,6,1,5,7,4,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [5,7,6,1,3,2,4] => [2,6,1,7,4,5,3] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [6,7,5,3,1,2,4] => [1,6,2,7,5,3,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [7,5,1,6,3,2,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [7,1,2,6,3,4,5] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [7,5,6,2,3,1,4] => [6,1,2,7,3,5,4] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [6,5,7,2,1,3,4] => [5,1,2,7,4,3,6] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [1,2,3,7,4,5,6] => 1 = 2 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,6,7,1,2,3,5] => [1,2,3,5,7,4,6] => 0 = 1 - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,6,1,7,2,4,5] => [4,1,3,2,7,5,6] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,6,7,3,4,5] => [3,2,1,4,7,5,6] => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,5,7,1,2,3,6] => [1,2,3,5,6,7,4] => 0 = 1 - 1
[(1,4),(2,3),(5,7),(6,8)]
=> [4,3,2,1,7,8,5,6] => [4,3,2,1,7,5,6] => [5,4,3,2,1,7,6] => 1 = 2 - 1
[(1,7),(2,5),(3,8),(4,6)]
=> [7,5,8,6,2,4,1,3] => [7,5,6,2,4,1,3] => [1,3,7,6,2,5,4] => 2 = 3 - 1
[(1,6),(2,7),(3,8),(4,5)]
=> [6,7,8,5,4,1,2,3] => [6,7,5,4,1,2,3] => [1,2,7,6,5,3,4] => 1 = 2 - 1
[(1,7),(2,8),(3,6),(4,5)]
=> [7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 1 = 2 - 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 0 = 1 - 1
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,9] => [7,6,8,5,4,3,2,1,9] => 1 = 2 - 1
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 1 = 2 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 0 = 1 - 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St001394
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[(1,2)]
=> [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => [4,3,2,1] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,2,3] => [1,3,4,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [4,3,2,1] => [1,4,3,2] => [2,1,3,4] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => [4,3,6,5,2,1] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,5,2,3,1,6] => [5,6,3,4,2,1] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => [6,5,4,3,2,1] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [6,4,3,1,2,5] => [1,6,5,3,4,2] => 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,4,3,6,5,2] => [2,5,4,1,3,6] => 1 = 2 - 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,5,6,3,4,2] => [2,6,1,5,3,4] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [6,5,1,3,2,4] => [1,2,4,6,5,3] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,6,2,1,3,5] => [5,1,4,3,6,2] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,2,6,1,4,5] => [4,3,1,5,6,2] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,6,5,4] => [4,3,2,1,5,6] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,2,6,5,3] => [5,2,3,1,4,6] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [5,1,6,2,4,3] => [6,2,1,4,3,5] => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,1,5,4,2,3] => [1,3,2,4,6,5] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => [2,1,3,4,5,6] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => [4,3,6,5,8,7,2,1] => 1 = 2 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => [5,6,3,4,8,7,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => [6,5,4,3,8,7,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [6,4,3,7,2,5,1,8] => [7,5,4,8,3,6,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => [8,5,4,7,6,3,2,1] => 1 = 2 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [8,4,3,6,5,1,2,7] => [1,6,5,8,7,3,4,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,4,3,6,5,8,7,2] => [2,5,4,7,6,1,3,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,5,6,3,4,8,7,2] => [2,6,7,4,5,1,3,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [8,5,6,3,4,1,2,7] => [1,7,8,5,6,3,4,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => [8,6,7,4,5,3,2,1] => 1 = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [6,5,7,3,2,4,1,8] => [7,6,8,4,3,5,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => [6,7,8,3,4,5,2,1] => 2 = 3 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => [5,7,3,8,4,6,2,1] => 1 = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => [4,3,7,8,5,6,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => [4,3,8,7,6,5,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,7,2,6,5,3,1,8] => [5,8,3,7,6,4,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [5,7,6,2,4,3,1,8] => [6,8,7,3,5,4,2,1] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => [7,8,6,5,3,4,2,1] => 2 = 3 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => 0 = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [8,6,5,4,3,1,2,7] => [1,8,7,6,5,3,4,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,6,5,4,3,8,7,2] => [2,7,6,5,4,1,3,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,7,5,4,8,3,6,2] => [2,8,6,5,1,7,3,4] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [8,7,5,4,1,3,2,6] => [1,2,8,7,4,6,5,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [7,8,5,4,1,2,3,6] => [8,1,7,6,3,4,5,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,8,5,4,2,1,3,7] => [7,1,8,6,4,3,5,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [5,8,6,2,4,1,3,7] => [6,1,8,4,7,3,5,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,8,2,6,5,1,3,7] => [5,1,4,8,7,3,6,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [3,2,8,6,5,1,4,7] => [4,3,1,8,7,5,6,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,6,5,8,7,4] => [4,3,2,7,6,1,5,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,1,2,6,5,8,7,3] => [5,2,3,7,6,1,4,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [5,1,6,2,4,8,7,3] => [6,2,7,3,5,1,4,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [6,1,5,4,2,8,7,3] => [7,2,6,5,3,1,4,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [7,1,5,4,8,2,6,3] => [8,2,6,5,1,4,3,7] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [8,1,5,4,7,6,2,3] => [1,3,7,6,2,4,8,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,8,5,4,7,6,3,2] => [2,1,7,6,3,4,5,8] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => [2,1,8,3,7,4,5,6] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [8,1,6,7,4,5,2,3] => [1,3,8,2,7,4,6,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [7,1,6,8,4,2,5,3] => [8,2,7,1,6,4,3,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [6,1,7,8,2,4,5,3] => [7,2,8,1,4,6,3,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [5,1,7,2,8,4,6,3] => [6,2,8,3,1,7,4,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [4,1,2,7,8,5,6,3] => [5,2,3,8,1,7,4,6] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [3,2,1,7,8,5,6,4] => [4,3,2,8,1,7,5,6] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [3,2,8,7,1,5,4,6] => [4,3,1,2,6,8,7,5] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [4,8,2,7,1,5,3,6] => [5,1,4,2,6,8,7,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [5,8,7,2,1,4,3,6] => [6,1,2,5,4,8,7,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [6,8,7,1,2,4,3,5] => [7,1,2,4,5,8,6,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [7,8,6,1,4,2,3,5] => [8,1,2,4,7,5,6,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [8,7,6,1,4,3,2,5] => [1,2,3,5,8,7,6,4] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,7,6,8,4,3,5,2] => [2,8,7,1,6,5,3,4] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,6,7,8,3,4,5,2] => [2,7,8,1,5,6,3,4] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [8,6,7,1,3,4,2,5] => [1,8,2,4,6,7,5,3] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [7,6,8,1,3,2,4,5] => [8,7,1,3,5,4,6,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [6,7,8,1,2,3,4,5] => [7,8,1,3,4,5,6,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [5,7,8,2,1,3,4,6] => [6,8,1,4,3,5,7,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [4,7,2,8,1,3,5,6] => [5,8,3,1,4,6,7,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [3,2,7,8,1,4,5,6] => [4,3,8,1,5,6,7,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [3,2,6,8,4,1,5,7] => [4,3,7,1,6,5,8,2] => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8} - 1
[(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => 0 = 1 - 1
[(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [3,2,1,8,7,6,5,4] => [4,3,2,1,5,6,7,8] => 0 = 1 - 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8] => 0 = 1 - 1
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => [10,9,8,7,6,5,4,3,2,1] => 0 = 1 - 1
[(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [7,6,5,4,3,2,1,10,9,8] => [8,7,6,5,4,3,2,1,9,10] => 0 = 1 - 1
[(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,10,9,8,7,6] => [6,5,4,3,2,1,7,8,9,10] => 0 = 1 - 1
[(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [3,2,1,10,9,8,7,6,5,4] => [4,3,2,1,5,6,7,8,9,10] => 0 = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10] => 0 = 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.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000862The number of parts of the shifted shape of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000001The number of reduced words for a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000233The number of nestings of a set partition. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000731The number of double exceedences of a permutation. St000402Half the size of the symmetry class of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000486The number of cycles of length at least 3 of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000141The maximum drop size of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000662The staircase size of the code of a permutation. St000359The number of occurrences of the pattern 23-1. St001153The number of blocks with even minimum in a set partition. St000872The number of very big descents of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!