***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000782 ----------------------------------------------------------------------------- Collection: Perfect matchings ----------------------------------------------------------------------------- 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]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$ ----------------------------------------------------------------------------- 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]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Created: Apr 18, 2017 at 16:13 by Manda Riehl ----------------------------------------------------------------------------- Last Updated: Apr 22, 2017 at 15:15 by Martin Rubey