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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001044
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(load all 11 compositions to match this statistic)
St001044: 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)]
=> 2
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 1
[(1,5),(2,4),(3,6)]
=> 1
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 2
Description
The number of pairs whose larger element is at most one more than half the size of the perfect matching.
Under the bijection between perfect matchings of {1,…,2n} and rooted unordered binary trees with n+1 labelled leaves described in Example 5.2.6 of [1], this is the number of nodes having two leaves as children.
The number of perfect matchings of {1,…,2n} with j such pairs, computed in [2], is
\frac{(2j-3)!! (n+1)!}{2^j j!}\binom{n-1}{2j-2}.
Matching statistic: St001553
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St000232
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}}
=> {{1,2,4},{3,6},{5}}
=> 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => {{1,3},{2,4},{5,6}}
=> {{1,4},{2,3,6},{5}}
=> 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => {{1,3},{2,4},{5,6}}
=> {{1,4},{2,3,6},{5}}
=> 1 = 2 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}}
=> {{1,2,6},{3,5},{4}}
=> 0 = 1 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => {{1,2},{3,5},{4,6}}
=> {{1,2,6},{3,5},{4}}
=> 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 0 = 1 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 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}}
=> {{1,2,4,8},{3,6},{5},{7}}
=> ? = 2 - 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}}
=> {{1,4,8},{2,3,6},{5},{7}}
=> ? = 2 - 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}}
=> {{1,4,8},{2,3,6},{5},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,8},{2,3,5},{4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,3,5},{4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 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}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,2,6},{3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,2,6},{3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,3,5},{4,8},{7}}
=> ? = 2 - 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}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => {{1,4},{2,5},{3,6},{7,8}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => {{1,4},{2,5},{3,6},{7,8}}
=> {{1,6},{2,5},{3,4,8},{7}}
=> ? = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => {{1,4},{2,5},{3,7},{6,8}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => {{1,4},{2,5},{3,7},{6,8}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => {{1,4},{2,5},{3,7},{6,8}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,3,5},{4,7},{6}}
=> ? = 2 - 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}}
=> {{1,2,8},{3,5},{4,7},{6}}
=> ? = 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}}
=> {{1,2,8},{3,5},{4,7},{6}}
=> ? = 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}}
=> {{1,8},{2,3,5},{4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => {{1,4},{2,5},{3,7},{6,8}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 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}}
=> {{1,8},{2,3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,8},{2,3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 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}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 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}}
=> {{1,8},{2,3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> ? = 1 - 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}}
=> {{1,2,8},{3,5},{4,7},{6}}
=> ? = 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}}
=> {{1,8},{2,3,5},{4,7},{6}}
=> ? = 2 - 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}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => {{1,4},{2,5},{3,7},{6,8}}
=> {{1,8},{2,5},{3,4,7},{6}}
=> ? = 2 - 1
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => {{1,4},{2,6},{3,7},{5,8}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1 - 1
[(1,5),(2,6),(3,8),(4,7)]
=> [5,6,7,8,1,2,4,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,8),(2,6),(3,7),(4,5)]
=> [5,6,7,8,4,2,3,1] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,7),(2,6),(3,8),(4,5)]
=> [5,6,7,8,4,2,1,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,7),(3,8),(4,5)]
=> [5,6,7,8,4,1,2,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,5),(2,7),(3,8),(4,6)]
=> [5,6,7,8,1,4,2,3] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> 0 = 1 - 1
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [6,7,8,9,10,1,2,3,4,5] => {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> 0 = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => {{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: St001605
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1 - 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? = 1 - 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? = 1 - 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? = 1 - 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? = 2 - 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? = 1 - 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? = 1 - 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? = 1 - 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? = 1 - 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? = 1 - 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? = 1 - 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? = 2 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> ? = 2 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> ? = 2 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> ? = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> ? = 2 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? = 2 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? = 2 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> ? = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? = 2 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? = 2 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> ? = 2 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? = 1 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? = 1 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? = 1 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? = 2 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? = 2 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? = 2 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2 = 3 - 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2 = 3 - 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,3),(2,8),(4,6),(5,7),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,7),(4,6),(5,8),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,8),(2,3),(4,6),(5,7),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,7),(2,3),(4,6),(5,8),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,6),(2,3),(4,7),(5,8),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,4),(2,3),(5,7),(6,8),(9,10)]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,4),(5,8),(6,7),(9,10)]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,6),(2,3),(4,8),(5,7),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,7),(2,3),(4,8),(5,6),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[(1,3),(2,6),(4,8),(5,7),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,7),(4,8),(5,6),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
[(1,3),(2,8),(4,7),(5,6),(9,10)]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1 = 2 - 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001394
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> [1,3,5,6,4,2] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> [2,1,5,6,4,3] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [[[.,.],.],[.,[[.,.],.]]]
=> [1,2,5,6,4,3] => 1 = 2 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [[[.,.],.],[[.,.],[.,.]]]
=> [1,2,4,6,5,3] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [[[.,.],.],[[.,.],[.,.]]]
=> [1,2,4,6,5,3] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,3,2,6,5,4] => 0 = 1 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,3,2,6,5,4] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [3,2,1,6,5,4] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [2,1,4,6,5,3] => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,4,3,6,5,2] => 0 = 1 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [[.,.],[[[.,.],.],[.,.]]]
=> [1,3,4,6,5,2] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [2,1,4,6,5,3] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [[.,[[.,.],.]],[.,[.,.]]]
=> [2,3,1,6,5,4] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [[[.,[.,.]],.],[.,[.,.]]]
=> [2,1,3,6,5,4] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,3,5,7,8,6,4,2] => 1 = 2 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [2,1,5,7,8,6,4,3] => ? = 2 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ? = 2 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> [1,2,4,7,8,6,5,3] => ? = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> [1,2,4,7,8,6,5,3] => ? = 2 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,2,4,6,8,7,5,3] => ? = 2 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,2,4,6,8,7,5,3] => ? = 2 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,3,2,6,8,7,5,4] => ? = 2 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,3,2,6,8,7,5,4] => ? = 2 - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [1,3,2,7,8,6,5,4] => ? = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [1,3,2,7,8,6,5,4] => ? = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [3,2,1,7,8,6,5,4] => ? = 2 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [2,1,4,7,8,6,5,3] => ? = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,4,3,7,8,6,5,2] => ? = 2 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> [1,3,4,7,8,6,5,2] => ? = 2 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [2,1,4,7,8,6,5,3] => ? = 2 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [[.,[[.,.],.]],[.,[.,[[.,.],.]]]]
=> [2,3,1,7,8,6,5,4] => ? = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [[[.,[.,.]],.],[.,[.,[[.,.],.]]]]
=> [2,1,3,7,8,6,5,4] => ? = 2 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => ? = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,2,3,6,8,7,5,4] => ? = 2 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,2,3,6,8,7,5,4] => ? = 2 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => ? = 1 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => ? = 1 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [[[.,[.,.]],.],[[.,.],[.,[.,.]]]]
=> [2,1,3,5,8,7,6,4] => ? = 1 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [[[.,[.,.]],.],[.,[[.,.],[.,.]]]]
=> [2,1,3,6,8,7,5,4] => ? = 2 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [[.,[[.,.],.]],[.,[[.,.],[.,.]]]]
=> [2,3,1,6,8,7,5,4] => ? = 2 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [2,1,4,6,8,7,5,3] => ? = 2 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,3,4,6,8,7,5,2] => ? = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,3,4,6,8,7,5,2] => ? = 2 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [2,1,4,6,8,7,5,3] => ? = 2 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [[.,[[.,.],.]],[.,[[.,.],[.,.]]]]
=> [2,3,1,6,8,7,5,4] => ? = 2 - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [[[.,[.,.]],.],[.,[[.,.],[.,.]]]]
=> [2,1,3,6,8,7,5,4] => ? = 2 - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [[[.,[.,.]],.],[[.,.],[.,[.,.]]]]
=> [2,1,3,5,8,7,6,4] => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [[[.,[.,.]],.],[[.,.],[.,[.,.]]]]
=> [2,1,3,5,8,7,6,4] => ? = 1 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,2,4,3,8,7,6,5] => ? = 1 - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 0 = 1 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 0 = 1 - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [2,4,3,1,8,7,6,5] => ? = 1 - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [[.,[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [2,3,1,5,8,7,6,4] => ? = 1 - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [2,1,5,4,8,7,6,3] => ? = 1 - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,3,5,4,8,7,6,2] => ? = 1 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,3,5,4,8,7,6,2] => ? = 1 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [2,1,5,4,8,7,6,3] => ? = 1 - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [[.,[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [2,3,1,5,8,7,6,4] => ? = 1 - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [2,4,3,1,8,7,6,5] => ? = 1 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 0 = 1 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,2,4,3,8,7,6,5] => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,2,4,3,8,7,6,5] => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,4,3,2,8,7,6,5] => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,4,3,2,8,7,6,5] => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,4,3,2,8,7,6,5] => ? = 1 - 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] => 0 = 1 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [3,2,1,5,8,7,6,4] => ? = 1 - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [2,1,5,4,8,7,6,3] => ? = 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.
Matching statistic: St000842
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00064: Permutations —reverse⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [5,6,2,1,4,3] => [2,6,4,1,5,3] => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [5,6,1,2,4,3] => [6,1,4,2,5,3] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [4,1,6,2,5,3] => [4,6,5,1,2,3] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,4,6,2,5,3] => [1,6,5,4,2,3] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,1,2,6,5,4] => [3,1,2,5,6,4] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [4,3,6,5,1,2] => [5,1,4,3,6,2] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,4,6,5,1,2] => [5,1,3,4,6,2] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [2,4,6,1,5,3] => [6,2,5,4,1,3] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [2,3,1,6,5,4] => [3,2,1,5,6,4] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 2 = 1 + 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] => [4,1,6,3,8,5,7,2] => ? = 2 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [7,8,5,6,2,1,4,3] => [2,6,4,1,8,5,7,3] => ? = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [7,8,5,6,1,2,4,3] => [6,1,4,2,8,5,7,3] => ? = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [7,8,4,1,6,2,5,3] => [4,8,6,5,3,2,7,1] => ? = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [7,8,1,4,6,2,5,3] => [8,6,5,1,4,3,7,2] => ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [6,1,8,4,7,2,5,3] => [6,7,5,8,2,1,4,3] => ? = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,6,8,4,7,2,5,3] => [1,7,5,8,2,6,4,3] => ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [1,6,8,3,2,7,5,4] => [1,3,8,5,7,6,2,4] => ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [6,1,8,3,2,7,5,4] => [6,3,8,5,7,1,2,4] => ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [7,8,1,3,2,6,5,4] => [8,3,1,5,6,2,7,4] => ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [7,8,3,1,2,6,5,4] => [3,1,8,5,6,2,7,4] => ? = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [7,8,3,2,1,6,5,4] => [2,3,8,5,6,1,7,4] => ? = 2 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [7,8,4,2,6,1,5,3] => [6,4,8,5,1,3,7,2] => ? = 2 + 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] => [5,1,4,6,8,2,7,3] => ? = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [7,8,3,4,6,5,1,2] => [5,1,6,3,8,2,7,4] => ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [7,8,2,4,6,1,5,3] => [6,8,5,2,1,4,7,3] => ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [7,8,2,3,1,6,5,4] => [3,8,2,5,6,1,7,4] => ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [7,8,2,1,3,6,5,4] => [2,8,1,5,6,3,7,4] => ? = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [7,8,1,2,3,6,5,4] => [8,1,2,5,6,3,7,4] => ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [6,1,8,2,3,7,5,4] => [6,8,2,5,7,1,3,4] => ? = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,6,8,2,3,7,5,4] => [1,8,2,5,7,6,3,4] => ? = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [1,5,2,8,3,7,6,4] => [1,5,8,6,2,7,3,4] => ? = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [5,1,2,8,3,7,6,4] => [5,1,8,6,2,7,3,4] => ? = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [5,2,1,8,3,7,6,4] => [2,5,8,6,1,7,3,4] => ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [6,2,8,1,3,7,5,4] => [8,6,1,5,7,2,3,4] => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [6,2,8,3,1,7,5,4] => [3,8,6,5,7,1,2,4] => ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [6,2,8,4,7,1,5,3] => [7,6,5,8,1,2,4,3] => ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [6,3,8,4,7,5,1,2] => [5,1,7,6,8,4,2,3] => ? = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [3,6,8,4,7,5,1,2] => [5,1,3,7,8,6,2,4] => ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [2,6,8,4,7,1,5,3] => [7,2,5,8,1,6,4,3] => ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [2,6,8,3,1,7,5,4] => [3,2,8,5,7,6,1,4] => ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [2,6,8,1,3,7,5,4] => [8,2,1,5,7,6,3,4] => ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [2,5,1,8,3,7,6,4] => [5,2,8,6,1,7,3,4] => ? = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [2,1,5,8,3,7,6,4] => [2,1,8,6,5,7,3,4] => ? = 1 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,2,5,8,3,7,6,4] => [1,2,8,6,5,7,3,4] => ? = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,2,4,3,6,7,8,5] => ? = 1 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [2,1,4,3,6,7,8,5] => ? = 1 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [2,4,1,3,8,7,6,5] => [4,2,1,3,6,7,8,5] => ? = 1 + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [2,4,3,1,8,7,6,5] => [3,2,4,1,6,7,8,5] => ? = 1 + 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [2,5,3,8,1,7,6,4] => [8,2,5,6,3,7,1,4] => ? = 1 + 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [2,5,4,8,7,1,6,3] => [7,2,6,5,4,1,8,3] => ? = 1 + 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [3,5,4,8,7,6,1,2] => [6,1,3,5,4,7,8,2] => ? = 1 + 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [5,3,4,8,7,6,1,2] => [6,1,5,3,4,7,8,2] => ? = 1 + 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [5,2,4,8,7,1,6,3] => [7,5,6,2,4,1,8,3] => ? = 1 + 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [5,2,3,8,1,7,6,4] => [8,5,2,6,3,7,1,4] => ? = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [4,2,3,1,8,7,6,5] => [3,4,2,1,6,7,8,5] => ? = 1 + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [4,2,1,3,8,7,6,5] => [2,4,1,3,6,7,8,5] => ? = 1 + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [4,1,2,3,6,7,8,5] => ? = 1 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,4,2,3,6,7,8,5] => ? = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,3,4,2,6,7,8,5] => ? = 1 + 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] => [2,3,4,1,6,7,8,5] => 2 = 1 + 1
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of \pi: \min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.
According to [1, Def.1.3], a permutation \pi is k-prolific, if the set of permutations obtained from \pi by deleting any k elements and standardising has maximal cardinality, i.e., \binom{n}{k}.
By [1, Thm.2.22], a permutation is k-prolific if and only if its breath is at least k+2.
By [1, Cor.4.3], the smallest permutations that are k-prolific have size \lceil k^2+2k+1\rceil, and by [1, Thm.4.4], there are k-prolific permutations of any size larger than this.
According to [2] the proportion of k-prolific permutations in the set of all permutations is asymptotically equal to \exp(-k^2-k).
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