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Statistic identifier: St000782

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Collection: Perfect matchings

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Description: The indicator function of whether a given perfect matching is an L & P matching. 

An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$,  insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.

The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$

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References: [1]   Jefferson, A. F. The substitution decomposition of matchings and RNA secondary structures [[MathSciNet:3439033]]
[2]   Saule, Cédric, Régnier, M., Steyaert, J.-M., Denise, A. Counting RNA pseudoknotted structures [[MathSciNet:2843853]]

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Code:


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Statistic values:

[(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)] => 1
[(1,6),(2,3),(4,5)] => 1
[(1,6),(2,4),(3,5)] => 0
[(1,5),(2,4),(3,6)] => 1
[(1,4),(2,5),(3,6)] => 0
[(1,3),(2,5),(4,6)] => 0
[(1,2),(3,5),(4,6)] => 1
[(1,2),(3,6),(4,5)] => 1
[(1,3),(2,6),(4,5)] => 1
[(1,4),(2,6),(3,5)] => 1
[(1,5),(2,6),(3,4)] => 1
[(1,6),(2,5),(3,4)] => 1

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Created: Apr 18, 2017 at 16:13 by Manda Riehl

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Last Updated: Apr 22, 2017 at 15:15 by Martin Rubey