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Your data matches 94 different statistics following compositions of up to 3 maps.
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Matching statistic: St000788
St000788: 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)]
=> 1
[(1,4),(2,3),(5,6)]
=> 1
[(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)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(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)]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 1
[(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)]
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> 4
[(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)]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> 1
[(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)]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> 1
[(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)]
=> 4
[(1,4),(2,7),(3,5),(6,8)]
=> 3
[(1,3),(2,7),(4,5),(6,8)]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> 4
Description
The number of nesting-similar perfect matchings of a perfect matching.
Consider the infinite tree T defined in [1] as follows. T has the perfect matchings on {1,…,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,…,2n} are nesting-similar, if the distribution of the number of nestings 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 nesting-similar, it is enough to check that M and N have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with n arcs as 2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}. [prop 3.6, 1] has further interpretations of this number.
Matching statistic: St001879
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> ([],1)
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ? ∊ {1,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ? ∊ {1,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ? ∊ {1,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000813
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 28%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 28%
Values
[(1,2)]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(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]
=> 10
[(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]
=> 10
[(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]
=> 9
[(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]
=> 9
[(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
[(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
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(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]
=> 9
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 3
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 3
[(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]
=> 9
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 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
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [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
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [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]
=> 4
[(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]
=> 4
[(1,5),(2,7),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,8),(2,5),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,7),(2,5),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,6),(2,5),(3,7),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [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]
=> 4
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
Description
The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.
Matching statistic: St000771
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 22%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 22%
Values
[(1,2)]
=> [1,0]
=> [1] => ([],1)
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2] => ([],2)
=> ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2] => ([],2)
=> ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0, 4.7\dots, 6, 9.2\dots and therefore statistic 1.
Matching statistic: St000772
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 22%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 22%
Values
[(1,2)]
=> [1,0]
=> [1] => ([],1)
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2] => ([],2)
=> ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2] => ([],2)
=> ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0, 4.7\dots, 6, 9.2\dots and therefore also statistic 1.
The graphs with statistic n-1, n-2 and n-3 have been characterised, see [1].
Matching statistic: St000264
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 9%●distinct values known / distinct values provided: 6%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 9%●distinct values known / distinct values provided: 6%
Values
[(1,2)]
=> [1,0]
=> [1] => ([],1)
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> ? ∊ {1,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,3),(5,9),(6,7),(8,10)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,9),(6,7),(8,10)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,10),(6,7),(8,9)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,5),(3,10),(6,8),(7,9)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,4),(3,10),(6,8),(7,9)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,3),(5,10),(6,8),(7,9)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,3),(2,4),(5,10),(6,8),(7,9)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,2),(3,4),(5,10),(6,8),(7,9)]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,2),(3,4),(5,9),(6,8),(7,10)]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,9),(6,8),(7,10)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,3),(5,9),(6,8),(7,10)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,4),(3,9),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,5),(3,9),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,5),(3,8),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,4),(3,8),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,3),(5,8),(6,9),(7,10)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,3),(2,4),(5,8),(6,9),(7,10)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,2),(3,4),(5,8),(6,9),(7,10)]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,4),(5,7),(6,9),(8,10)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,3),(5,7),(6,9),(8,10)]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,5),(2,10),(3,4),(6,8),(7,9)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,10),(3,5),(6,8),(7,9)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,9),(3,5),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,9),(3,4),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,8),(3,4),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,8),(3,5),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,10),(2,5),(3,4),(6,8),(7,9)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,9),(2,5),(3,4),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,8),(2,5),(3,4),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,6),(2,5),(3,4),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,6),(3,4),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,4),(2,6),(3,5),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,3),(2,6),(4,5),(7,9),(8,10)]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,3),(2,5),(4,6),(7,9),(8,10)]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[(1,4),(2,5),(3,6),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,5),(2,4),(3,6),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,6),(2,4),(3,5),(7,9),(8,10)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,8),(2,4),(3,5),(6,9),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[(1,9),(2,4),(3,5),(6,8),(7,10)]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001087
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
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] => [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [3,4,2,1] => [3,4,2,1] => [4,1,3,2] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => [5,6,4,1,3,2] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,6,2,5,1,4] => [6,4,5,1,3,2] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,6,2,5,4,1] => [5,4,6,1,3,2] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [4,6,5,2,3,1] => [5,3,6,1,4,2] => 0 = 1 - 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [4,6,5,2,1,3] => [6,3,5,1,4,2] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1 = 2 - 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [4,6,5,3,1,2] => [5,1,4,3,6,2] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,6,5,3,2,1] => [6,1,4,3,5,2] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => [6,7,5,8,4,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [3,8,2,7,1,6,5,4] => [6,8,4,7,5,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => [8,4,7,5,6,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,8,2,7,6,5,1,4] => [6,5,8,4,7,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,8,2,7,6,5,4,1] => [6,5,7,4,8,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,8,7,2,6,5,3,1] => [6,5,7,3,8,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,8,7,2,6,5,1,3] => [6,5,8,3,7,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,8,7,2,6,1,5,3] => [8,3,7,5,6,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,8,7,2,1,6,5,3] => [6,8,3,7,5,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,8,7,3,1,6,5,2] => [6,7,5,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => [7,6,1,4,3,8,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,8,7,3,2,6,1,5] => [6,7,1,4,3,8,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,8,7,3,2,6,5,1] => [6,8,1,4,3,7,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,8,7,3,6,2,5,1] => [8,1,4,3,7,5,6,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,8,7,3,6,2,1,5] => [7,1,4,3,8,5,6,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,8,7,3,6,1,5,2] => [7,5,6,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,8,7,3,1,6,5,2] => [6,7,5,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [4,8,7,3,1,6,5,2] => [6,7,5,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [4,8,7,3,6,1,5,2] => [7,5,6,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [4,8,7,3,6,5,1,2] => [6,5,7,1,4,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => [6,5,8,1,4,3,7,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [5,8,7,6,3,4,2,1] => [6,4,8,1,5,3,7,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => [6,4,7,1,5,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [5,8,7,6,3,1,4,2] => [7,4,6,1,5,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 0 = 1 - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 0 = 1 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [5,8,7,6,3,1,4,2] => [7,4,6,1,5,3,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [5,8,7,6,3,2,1,4] => [7,1,5,3,8,4,6,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [5,8,7,6,3,2,4,1] => [8,1,5,3,7,4,6,2] => 0 = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [5,8,7,6,2,4,3,1] => [6,4,7,3,8,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [5,8,7,6,2,4,1,3] => [6,4,8,3,7,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [5,8,7,6,2,1,4,3] => [8,3,7,4,6,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 0 = 1 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 2 = 3 - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1 = 2 - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [4,8,7,2,1,6,5,3] => [6,8,3,7,5,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [4,8,7,2,6,1,5,3] => [8,3,7,5,6,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [4,8,7,2,6,5,1,3] => [6,5,8,3,7,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [4,8,7,2,6,5,3,1] => [6,5,7,3,8,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,6),(5,7)]
=> [3,6,2,7,8,4,5,1] => [3,8,2,7,6,5,4,1] => [6,5,7,4,8,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,3),(4,6),(5,8)]
=> [3,6,2,7,8,4,1,5] => [3,8,2,7,6,5,1,4] => [6,5,8,4,7,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,7),(5,8)]
=> [3,6,2,7,8,1,4,5] => [3,8,2,7,6,1,5,4] => [8,4,7,5,6,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => [3,8,2,7,1,6,5,4] => [6,8,4,7,5,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [3,8,2,1,7,6,5,4] => [6,7,5,8,4,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 2 = 3 - 1
[(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => [6,7,5,8,4,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,3),(4,8),(6,7)]
=> [3,5,2,7,1,8,6,4] => [3,8,2,7,1,6,5,4] => [6,8,4,7,5,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,8),(5,7)]
=> [3,6,2,7,8,1,5,4] => [3,8,2,7,6,1,5,4] => [8,4,7,5,6,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,3),(4,8),(5,6)]
=> [3,6,2,7,8,5,1,4] => [3,8,2,7,6,5,1,4] => [6,5,8,4,7,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [3,8,2,7,6,5,4,1] => [6,5,7,4,8,1,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => [4,8,7,2,6,5,3,1] => [6,5,7,3,8,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,4),(3,8),(5,6)]
=> [4,6,7,2,8,5,1,3] => [4,8,7,2,6,5,1,3] => [6,5,8,3,7,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,8),(5,7)]
=> [4,6,7,2,8,1,5,3] => [4,8,7,2,6,1,5,3] => [8,3,7,5,6,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,4),(3,8),(6,7)]
=> [4,5,7,2,1,8,6,3] => [4,8,7,2,1,6,5,3] => [6,8,3,7,5,1,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [5,8,7,6,2,1,4,3] => [8,3,7,4,6,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => [5,8,7,6,2,4,1,3] => [6,4,8,3,7,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => [5,8,7,6,2,4,3,1] => [6,4,7,3,8,1,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern 123, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St000891
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 11%
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 11%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [3,4,1,2] => [4,3,2,1] => 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)]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [5,6,2,1,4,3] => [6,5,4,3,2,1] => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,6,1,2,3,4] => [6,5,3,4,2,1] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,1,6,2,3,5] => [5,2,6,4,1,3] => 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => 3 = 2 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [4,3,6,5,1,2] => [6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,4,5,6,1,2] => [6,5,3,4,2,1] => 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [2,4,5,1,6,3] => [4,6,3,1,5,2] => 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 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] => [8,7,6,5,4,3,2,1] => 2 = 1 + 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] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [7,8,5,6,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [7,8,4,1,6,2,3,5] => [8,7,6,4,5,3,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,8,1,4,5,2,3,6] => [8,7,3,6,5,4,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [6,1,8,4,5,2,3,7] => [7,2,8,6,5,4,1,3] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,6,7,4,5,2,3,8] => [1,7,6,5,4,3,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,6,7,3,2,5,4,8] => [1,7,6,5,4,3,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [6,1,8,3,2,5,4,7] => [7,2,8,5,4,6,1,3] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,8,1,3,2,5,4,6] => [8,7,3,5,4,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [7,8,3,1,2,6,4,5] => [8,7,5,4,3,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 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] => [8,7,5,4,3,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 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] => [8,7,6,5,4,3,2,1] => 2 = 1 + 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] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [7,8,2,4,5,1,6,3] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [7,8,2,3,1,5,6,4] => [8,7,5,4,3,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [7,8,2,1,3,4,6,5] => [8,7,4,3,5,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,8,1,2,3,4,5,6] => [8,7,3,4,5,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [6,1,8,2,3,4,5,7] => [7,2,8,4,5,6,1,3] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,6,7,2,3,4,5,8] => [1,7,6,4,5,3,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,5,2,7,3,4,6,8] => [1,6,3,7,5,2,4,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [5,1,2,8,3,4,6,7] => [6,2,3,8,5,1,7,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [5,2,1,8,3,4,7,6] => [6,3,2,8,5,1,7,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,2,8,1,3,4,7,5] => [8,4,7,2,5,6,3,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [6,2,8,3,1,5,7,4] => [8,5,7,4,2,6,3,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [6,2,8,4,5,1,7,3] => [8,6,7,5,4,2,3,1] => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [6,3,8,4,5,7,1,2] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,6,7,4,5,8,1,2] => [8,7,5,4,3,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [2,6,7,4,5,1,8,3] => [6,8,5,4,3,1,7,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [2,6,7,3,1,5,8,4] => [5,8,6,4,1,3,7,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [2,6,7,1,3,4,8,5] => [4,8,6,1,5,3,7,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [2,5,1,7,3,4,8,6] => [3,6,1,8,5,2,7,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => 3 = 2 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,5,6,3,4,7,8] => [1,2,6,5,4,3,7,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => 3 = 2 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [2,4,1,3,7,5,8,6] => [3,4,1,2,8,6,7,5] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [2,4,3,1,7,6,8,5] => [4,3,2,1,8,6,7,5] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [2,5,3,7,1,6,8,4] => [5,8,3,6,1,4,7,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [2,5,4,7,6,1,8,3] => [6,8,3,5,4,1,7,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [3,5,4,7,6,8,1,2] => [8,7,3,5,4,6,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [5,3,4,8,6,7,1,2] => [8,7,3,6,5,4,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [5,2,4,8,6,1,7,3] => [8,6,3,7,5,2,4,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [5,2,3,8,1,6,7,4] => [8,5,3,7,2,6,4,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [4,2,3,1,8,6,7,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [4,3,2,1,8,6,7,5] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [4,2,3,1,8,6,7,5] => 3 = 2 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,4,3,2,7,6,5,8] => 3 = 2 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5,8] => 3 = 2 + 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [4,2,3,1,8,7,6,5] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [4,3,2,1,8,7,6,5] => 2 = 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] => [4,3,2,1,8,7,6,5] => 2 = 1 + 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] => [8,5,3,7,2,6,4,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [5,4,2,8,7,1,6,3] => [8,6,3,7,5,2,4,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [5,4,3,8,7,6,1,2] => [8,7,3,6,5,4,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [6,4,8,3,7,5,1,2] => [8,7,6,5,4,3,2,1] => 2 = 1 + 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] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [6,3,8,2,1,7,5,4] => [8,5,7,4,2,6,3,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,4),(3,7),(6,8)]
=> [5,4,7,2,1,8,3,6] => [6,3,8,1,2,7,4,5] => [8,5,7,4,2,6,3,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,4),(3,7),(5,8)]
=> [6,4,7,2,8,1,3,5] => [5,3,1,8,2,7,4,6] => [7,5,3,8,2,6,1,4] => 3 = 2 + 1
[(1,7),(2,4),(3,6),(5,8)]
=> [7,4,6,2,8,3,1,5] => [5,1,3,8,2,6,4,7] => [7,2,5,8,3,6,1,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [1,5,3,7,2,6,4,8] => [1,7,5,6,3,4,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [1,5,4,7,6,2,3,8] => [1,7,6,5,4,3,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,3),(4,6),(5,8)]
=> [7,3,2,6,8,4,1,5] => [5,1,4,8,6,2,3,7] => [7,2,6,8,5,3,1,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,3),(4,7),(5,8)]
=> [6,3,2,7,8,1,4,5] => [5,4,1,8,7,2,3,6] => [7,6,3,8,5,2,1,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,3),(4,7),(6,8)]
=> [5,3,2,7,1,8,4,6] => [6,4,8,1,7,2,3,5] => [8,7,6,4,5,3,2,1] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,4),(2,3),(5,7),(6,8)]
=> [4,3,2,1,7,8,5,6] => [6,5,8,7,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [6,5,8,7,2,1,4,3] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [6,5,8,7,3,4,1,2] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [5,6,7,8,3,4,1,2] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,3),(2,4),(5,8),(6,7)]
=> [3,4,1,2,8,7,6,5] => [5,6,7,8,2,1,4,3] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => 2 = 1 + 1
[(1,5),(2,3),(4,8),(6,7)]
=> [5,3,2,8,1,7,6,4] => [4,6,7,1,8,2,3,5] => [7,8,6,4,5,3,1,2] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,6),(2,3),(4,8),(5,7)]
=> [6,3,2,8,7,1,5,4] => [4,5,1,7,8,2,3,6] => [7,6,3,8,5,2,1,4] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,7),(2,3),(4,8),(5,6)]
=> [7,3,2,8,6,5,1,4] => [4,1,5,6,8,2,3,7] => [7,2,6,4,8,3,1,5] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,4,5,6,7,2,3,8] => [1,7,6,4,5,3,2,8] => ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} + 1
[(1,5),(2,6),(3,8),(4,7)]
=> [5,6,8,7,1,2,4,3] => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 1
[(1,6),(2,5),(3,8),(4,7)]
=> [6,5,8,7,2,1,4,3] => [3,4,1,2,7,8,5,6] => [4,3,2,1,8,7,6,5] => 2 = 1 + 1
[(1,8),(2,5),(3,7),(4,6)]
=> [8,5,7,6,2,4,3,1] => [1,3,4,2,6,7,5,8] => [1,4,3,2,7,6,5,8] => 3 = 2 + 1
[(1,8),(2,6),(3,7),(4,5)]
=> [8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => 3 = 2 + 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: St000648
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[(1,2)]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [3,2,4,1] => [4,2,1,3] => 0 = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [4,3,5,2,6,1] => [6,4,2,1,3,5] => 1 = 2 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,5,3,4,6,1] => [6,1,3,4,2,5] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,6,1] => [6,4,3,2,1,5] => 1 = 2 - 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [1,5,4,6,2,3] => [1,5,6,3,2,4] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,4,2,5,3,1] => [6,3,5,2,4,1] => 1 = 2 - 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [6,1,5,2,4,3] => [2,4,6,5,3,1] => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [5,1,6,3,2,4] => [2,5,4,6,1,3] => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,1,3,6,2,5] => [2,5,3,1,6,4] => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,2,1,6,4,5] => [3,2,1,5,6,4] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [5,4,6,3,2,1] => [6,5,4,2,1,3] => 0 = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,6,4,5,2,1] => [6,5,1,3,4,2] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [2,6,5,3,4,1] => [6,1,4,5,3,2] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [1,6,5,4,2,3] => [1,5,6,4,3,2] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [5,4,6,3,7,2,8,1] => [8,6,4,2,1,3,5,7] => 0 = 1 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,6,4,5,7,2,8,1] => [8,6,1,3,4,2,5,7] => 0 = 1 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [6,5,4,3,7,2,8,1] => [8,6,4,3,2,1,5,7] => 0 = 1 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [2,6,5,7,3,4,8,1] => [8,1,5,6,3,2,4,7] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,5,3,6,4,2,8,1] => [8,6,3,5,2,4,1,7] => 0 = 1 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [1,6,4,7,5,8,2,3] => [1,7,8,3,5,2,4,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [8,5,3,6,2,7,4,1] => [8,5,3,7,2,4,6,1] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [8,2,6,3,4,7,5,1] => [8,2,4,5,7,3,6,1] => 1 = 2 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [1,3,7,4,6,8,2,5] => [1,7,2,4,8,5,3,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,2,6,3,5,4,8,1] => [8,2,4,6,5,3,1,7] => 1 = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [6,2,7,4,3,5,8,1] => [8,2,5,4,6,1,3,7] => 1 = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,3,7,4,5,6,8,1] => [8,1,2,4,5,6,3,7] => 0 = 1 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [5,2,4,7,3,6,8,1] => [8,2,5,3,1,6,4,7] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,3,2,7,5,6,8,1] => [8,3,2,1,5,6,4,7] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [6,5,7,4,3,2,8,1] => [8,6,5,4,2,1,3,7] => 1 = 2 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,7,5,6,3,2,8,1] => [8,6,5,1,3,4,2,7] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [3,7,6,4,5,2,8,1] => [8,6,1,4,5,3,2,7] => 0 = 1 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [2,7,6,5,3,4,8,1] => [8,1,5,6,4,3,2,7] => 2 = 3 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => 1 = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [1,7,6,5,4,8,2,3] => [1,7,8,5,4,3,2,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [8,6,5,4,2,7,3,1] => [8,5,7,4,3,2,6,1] => 0 = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [8,1,6,5,7,2,4,3] => [2,6,8,7,4,3,5,1] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [7,1,6,5,8,3,2,4] => [2,7,6,8,4,3,1,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [1,2,7,6,8,3,4,5] => [1,2,6,7,8,4,3,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,1,7,5,3,8,2,4] => [2,7,5,8,4,1,3,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [2,1,7,5,6,8,3,4] => [2,1,7,8,4,5,3,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,6,7,5,8,2,4] => [2,7,1,8,5,3,4,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [5,4,1,7,6,8,2,3] => [3,7,8,2,1,5,4,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [6,5,8,4,2,7,3,1] => [8,5,7,4,2,1,6,3] => 0 = 1 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,8,5,6,2,7,3,1] => [8,5,7,1,3,4,6,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [3,8,6,4,2,7,5,1] => [8,5,1,4,7,3,6,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [7,8,5,2,4,6,3,1] => [8,4,7,5,3,6,1,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [2,8,6,3,7,4,5,1] => [8,1,4,6,7,3,5,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [1,8,6,3,7,5,2,4] => [1,7,4,8,6,3,5,2] => 0 = 1 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,7,5,2,6,4,3,1] => [8,4,7,6,3,5,2,1] => 2 = 3 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [8,7,1,6,2,5,4,3] => [3,5,8,7,6,4,2,1] => 1 = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => 1 = 2 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [7,8,1,6,2,4,5,3] => [3,5,8,6,7,4,1,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [2,8,1,7,4,3,6,5] => [3,1,6,5,8,7,4,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [6,8,1,4,7,2,5,3] => [3,6,8,4,7,1,5,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [3,8,4,1,7,2,6,5] => [4,6,1,3,8,7,5,2] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [5,4,8,1,7,2,6,3] => [4,6,8,2,1,7,5,3] => 0 = 1 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [5,4,7,1,8,3,2,6] => [4,7,6,2,1,8,3,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [3,7,4,1,8,5,2,6] => [4,7,1,3,6,8,2,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [6,7,1,4,8,3,2,5] => [3,7,6,4,8,1,2,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [2,7,1,8,4,5,3,6] => [3,1,7,5,6,8,2,4] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [1,7,2,8,5,3,4,6] => [1,3,6,7,5,8,2,4] => 1 = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [7,6,1,8,4,3,2,5] => [3,7,6,5,8,2,1,4] => 1 = 2 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [8,6,1,7,4,2,5,3] => [3,6,8,5,7,2,4,1] => 2 = 3 - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [8,1,2,7,3,4,6,5] => [2,3,5,6,8,7,4,1] => 2 = 3 - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [7,1,2,8,3,5,4,6] => [2,3,5,7,6,8,1,4] => 2 = 3 - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [6,1,2,8,4,3,5,7] => [2,3,6,5,7,1,8,4] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => [1,2,3,5,6,7,8,4] => 0 = 1 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [5,1,2,4,8,3,6,7] => [2,3,6,4,1,7,8,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [4,1,3,2,8,5,6,7] => [2,4,3,1,6,7,8,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [3,2,1,4,8,5,6,7] => [3,2,1,4,6,7,8,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [4,3,6,1,5,8,2,7] => [4,7,2,1,5,3,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [5,6,2,1,4,8,3,7] => [4,3,7,5,1,2,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [2,6,1,4,5,8,3,7] => [3,1,7,4,5,2,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [5,4,7,2,1,8,3,6] => [6,5,1,4,3,8,2,7] => [3,7,5,4,2,1,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,7),(5,8)]
=> [6,4,7,2,8,1,3,5] => [1,6,2,5,8,3,4,7] => [1,3,6,7,4,2,8,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,4),(3,6),(5,8)]
=> [7,4,6,2,8,3,1,5] => [7,5,1,3,8,4,2,6] => [3,7,4,6,2,8,1,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [8,5,1,3,7,2,6,4] => [3,6,4,8,2,7,5,1] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [8,4,2,1,7,3,6,5] => [4,3,6,2,8,7,5,1] => 0 = 1 - 1
[(1,7),(2,3),(4,6),(5,8)]
=> [7,3,2,6,8,4,1,5] => [7,4,2,1,8,5,3,6] => [4,3,7,2,6,8,1,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,7),(5,8)]
=> [6,3,2,7,8,1,4,5] => [1,5,4,2,8,3,6,7] => [1,4,6,3,2,7,8,5] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,5),(2,3),(4,7),(6,8)]
=> [5,3,2,7,1,8,4,6] => [6,4,3,1,5,8,2,7] => [4,7,3,2,5,1,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,4),(2,3),(5,7),(6,8)]
=> [4,3,2,1,7,8,5,6] => [5,4,3,2,1,8,6,7] => [5,4,3,2,1,7,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [2,5,3,4,1,8,6,7] => [5,1,3,4,2,7,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [4,3,5,2,1,8,6,7] => [5,4,2,1,3,7,8,6] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => 0 = 1 - 1
[(1,3),(2,4),(5,8),(6,7)]
=> [3,4,1,2,8,7,6,5] => [4,7,5,6,8,3,2,1] => [8,7,6,1,3,4,2,5] => 0 = 1 - 1
[(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => 0 = 1 - 1
[(1,5),(2,3),(4,8),(6,7)]
=> [5,3,2,8,1,7,6,4] => [3,7,6,8,4,5,2,1] => [8,7,1,5,6,3,2,4] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,3),(4,8),(5,7)]
=> [6,3,2,8,7,1,5,4] => [2,7,6,8,5,3,4,1] => [8,1,6,7,5,3,2,4] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,3),(4,8),(5,6)]
=> [7,3,2,8,6,5,1,4] => [1,7,6,8,5,4,2,3] => [1,7,8,6,5,3,2,4] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [8,6,4,7,5,3,2,1] => [8,7,6,3,5,2,4,1] => 0 = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [8,3,7,4,6,5,2,1] => [8,7,2,4,6,5,3,1] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,7),(2,4),(3,8),(5,6)]
=> [7,4,8,2,6,5,1,3] => [1,4,8,6,7,5,2,3] => [1,7,8,2,6,4,5,3] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
[(1,6),(2,4),(3,8),(5,7)]
=> [6,4,8,2,7,1,5,3] => [2,4,8,6,7,3,5,1] => [8,1,6,2,7,4,5,3] => 1 = 2 - 1
[(1,5),(2,4),(3,8),(6,7)]
=> [5,4,8,2,1,7,6,3] => [7,3,8,5,4,6,2,1] => [8,7,2,5,4,6,1,3] => ? ∊ {1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 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: St000035
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[(1,2)]
=> [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 2
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 2
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 2
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 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] => [2,4,6,8,1,3,5,7] => 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,3,2,1,6,5,8,7] => [4,3,2,6,8,1,5,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [4,3,2,6,8,1,5,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [5,6,3,4,1,2,8,7] => [5,3,1,6,4,8,2,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,5,3,4,2,1,8,7] => [3,6,5,4,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => [3,5,8,2,7,6,4,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [8,5,7,4,2,6,3,1] => [2,8,5,4,7,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [5,6,3,4,1,2,8,7] => [5,3,1,6,4,8,2,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => [6,5,2,4,8,1,3,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => [6,5,2,4,8,1,3,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [6,5,3,4,2,1,8,7] => [3,6,5,4,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => [2,8,5,4,7,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,7,8,5,6,3,4] => [7,5,8,6,2,1,3,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => [5,8,7,6,2,4,1,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [8,5,3,7,2,6,4,1] => [3,5,8,2,7,6,4,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [8,5,7,4,2,6,3,1] => [2,8,5,4,7,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [8,5,7,4,2,6,3,1] => [2,8,5,4,7,6,3,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [8,7,3,6,5,4,2,1] => [8,7,3,6,5,4,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => [8,7,6,5,2,4,1,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => [8,7,6,5,2,4,1,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [7,8,3,6,5,4,1,2] => [7,8,3,1,6,5,4,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [6,8,3,7,5,1,4,2] => [3,6,8,1,7,5,4,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [8,7,6,5,2,4,1,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,7,8,5,6,3,4] => [7,5,8,6,2,1,3,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [5,7,3,8,1,6,2,4] => [5,7,3,1,8,2,6,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [7,5,8,4,2,6,1,3] => [5,2,7,4,1,8,6,3] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [7,8,6,4,5,3,1,2] => [7,4,1,8,6,5,3,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [8,7,6,4,5,3,2,1] => [4,8,7,6,5,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,8),(2,3),(4,6),(5,7)]
=> [3,6,2,7,8,4,5,1] => [8,7,3,6,5,4,2,1] => [8,7,3,6,5,4,2,1] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,7),(2,3),(4,6),(5,8)]
=> [3,6,2,7,8,4,1,5] => [7,8,3,6,5,4,1,2] => [7,8,3,1,6,5,4,2] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6}
[(1,5),(2,6),(3,8),(4,7)]
=> [5,6,7,8,1,2,4,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,6),(3,7),(4,5)]
=> [5,6,7,8,4,2,3,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,6),(3,8),(4,5)]
=> [5,6,7,8,4,2,1,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,7),(3,8),(4,5)]
=> [5,6,7,8,4,1,2,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,7),(3,8),(4,6)]
=> [5,6,7,8,1,4,2,3] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 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] => [2,4,6,8,10,1,3,5,7,9] => 1
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [6,7,8,9,10,4,5,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [6,7,8,9,10,1,4,5,3,2] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [6,7,8,9,10,1,4,5,2,3] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
Description
The number of left outer peaks of a permutation.
A left 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 1 if w_1 > w_2.
In other words, it is a peak in the word [0,w_1,..., w_n].
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000352The Elizalde-Pak rank of a permutation. St000359The number of occurrences of the pattern 23-1. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000862The number of parts of the shifted shape of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000647The number of big descents of a permutation. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001115The number of even descents of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000232The number of crossings of a set partition. St001394The genus of a permutation. St000233The number of nestings of a set partition. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000871The number of very big ascents of a permutation. St000834The number of right outer peaks of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000422The energy of a graph, if it is integral. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000254The nesting number of a set partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000732The number of double deficiencies of a permutation. St000402Half the size of the symmetry class of a permutation. St000665The number of rafts of a permutation. St000883The number of longest increasing subsequences of a permutation. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000909The number of maximal chains of maximal size in a poset. St001732The number of peaks visible from the left. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000523The number of 2-protected nodes of a rooted tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000441The number of successions of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000659The number of rises of length at least 2 of a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000872The number of very big descents of a permutation. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St001434The number of negative sum pairs of a signed permutation. St001549The number of restricted non-inversions between exceedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000570The Edelman-Greene number of a permutation. St001513The number of nested exceedences of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000650The number of 3-rises of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St000664The number of right ropes of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001947The number of ties in a parking function. St000052The number of valleys of a Dyck path not on the x-axis. St001728The number of invisible descents of a permutation.
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