Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000788
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Mapping
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 $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$. Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of 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.