***************************************************************************** * 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: St000043 ----------------------------------------------------------------------------- Collection: Perfect matchings ----------------------------------------------------------------------------- Description: The number of crossings plus two-nestings of a perfect matching. This is $C+2N$ where $C$ is the number of crossings ([[St000042]]) and $N$ is the number of nestings ([[St000041]]). The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)]. ----------------------------------------------------------------------------- References: [1] Simion, R., Stanton, D. Octabasic Laguerre polynomials and permutation statistics [[MathSciNet:1418763]] ----------------------------------------------------------------------------- Code: def statistic(x): return len(x.crossings()) + 2*len(x.nestings()) ----------------------------------------------------------------------------- Statistic values: [(1,2)] => 0 [(1,2),(3,4)] => 0 [(1,3),(2,4)] => 1 [(1,4),(2,3)] => 2 [(1,2),(3,4),(5,6)] => 0 [(1,3),(2,4),(5,6)] => 1 [(1,4),(2,3),(5,6)] => 2 [(1,5),(2,3),(4,6)] => 3 [(1,6),(2,3),(4,5)] => 4 [(1,6),(2,4),(3,5)] => 5 [(1,5),(2,4),(3,6)] => 4 [(1,4),(2,5),(3,6)] => 3 [(1,3),(2,5),(4,6)] => 2 [(1,2),(3,5),(4,6)] => 1 [(1,2),(3,6),(4,5)] => 2 [(1,3),(2,6),(4,5)] => 3 [(1,4),(2,6),(3,5)] => 4 [(1,5),(2,6),(3,4)] => 5 [(1,6),(2,5),(3,4)] => 6 [(1,8),(2,3),(4,5),(6,7)] => 6 [(1,8),(2,4),(3,5),(6,7)] => 7 [(1,8),(2,5),(3,4),(6,7)] => 8 [(1,8),(2,6),(3,4),(5,7)] => 9 [(1,2),(3,8),(4,5),(6,7)] => 4 [(1,3),(2,8),(4,5),(6,7)] => 5 [(1,4),(2,8),(3,5),(6,7)] => 6 [(1,5),(2,8),(3,4),(6,7)] => 7 [(1,6),(2,8),(3,4),(5,7)] => 8 [(1,7),(2,8),(3,4),(5,6)] => 9 [(1,8),(2,7),(3,4),(5,6)] => 10 [(1,8),(2,7),(3,5),(4,6)] => 11 [(1,7),(2,8),(3,5),(4,6)] => 10 [(1,6),(2,8),(3,5),(4,7)] => 9 [(1,5),(2,8),(3,6),(4,7)] => 8 [(1,4),(2,8),(3,6),(5,7)] => 7 [(1,3),(2,8),(4,6),(5,7)] => 6 [(1,2),(3,8),(4,6),(5,7)] => 5 [(1,8),(2,6),(3,5),(4,7)] => 10 [(1,8),(2,5),(3,6),(4,7)] => 9 [(1,8),(2,4),(3,6),(5,7)] => 8 [(1,8),(2,3),(4,6),(5,7)] => 7 [(1,8),(2,3),(4,7),(5,6)] => 8 [(1,8),(2,4),(3,7),(5,6)] => 9 [(1,8),(2,5),(3,7),(4,6)] => 10 [(1,8),(2,6),(3,7),(4,5)] => 11 [(1,2),(3,8),(4,7),(5,6)] => 6 [(1,3),(2,8),(4,7),(5,6)] => 7 [(1,4),(2,8),(3,7),(5,6)] => 8 [(1,5),(2,8),(3,7),(4,6)] => 9 [(1,6),(2,8),(3,7),(4,5)] => 10 [(1,7),(2,8),(3,6),(4,5)] => 11 [(1,8),(2,7),(3,6),(4,5)] => 12 [(1,2),(3,4),(8,5),(6,7)] => 2 [(1,2),(3,4),(8,6),(5,7)] => 1 [(1,2),(3,4),(8,7),(5,6)] => 0 [(1,2),(3,5),(8,4),(6,7)] => 3 [(1,2),(3,5),(8,6),(4,7)] => 2 [(1,2),(3,5),(8,7),(4,6)] => 1 [(1,2),(3,6),(8,4),(5,7)] => 4 [(1,2),(3,6),(8,5),(4,7)] => 3 [(1,2),(3,6),(8,7),(4,5)] => 2 [(1,2),(3,7),(8,4),(5,6)] => 5 [(1,2),(3,7),(8,5),(4,6)] => 4 [(1,2),(3,7),(8,6),(4,5)] => 3 [(1,3),(2,4),(8,5),(6,7)] => 3 [(1,3),(2,4),(8,6),(5,7)] => 2 [(1,3),(2,4),(8,7),(5,6)] => 1 [(1,3),(2,5),(8,4),(6,7)] => 4 [(1,3),(2,5),(8,6),(4,7)] => 3 [(1,3),(2,5),(8,7),(4,6)] => 2 [(1,3),(2,6),(8,4),(5,7)] => 5 [(1,3),(2,6),(8,5),(4,7)] => 4 [(1,3),(2,6),(8,7),(4,5)] => 3 [(1,3),(2,7),(8,4),(5,6)] => 6 [(1,3),(2,7),(8,5),(4,6)] => 5 [(1,3),(2,7),(8,6),(4,5)] => 4 [(1,4),(2,3),(8,5),(6,7)] => 4 [(1,4),(2,3),(8,6),(5,7)] => 3 [(1,4),(2,3),(8,7),(5,6)] => 2 [(1,4),(2,5),(8,3),(6,7)] => 5 [(1,4),(2,5),(8,6),(3,7)] => 4 [(1,4),(2,5),(8,7),(3,6)] => 3 [(1,4),(2,6),(8,3),(5,7)] => 6 [(1,4),(2,6),(8,5),(3,7)] => 5 [(1,4),(2,6),(8,7),(3,5)] => 4 [(1,4),(2,7),(8,3),(5,6)] => 7 [(1,4),(2,7),(8,5),(3,6)] => 6 [(1,4),(2,7),(8,6),(3,5)] => 5 [(1,5),(2,3),(8,4),(6,7)] => 5 [(1,5),(2,3),(8,6),(4,7)] => 4 [(1,5),(2,3),(8,7),(4,6)] => 3 [(1,5),(2,4),(8,3),(6,7)] => 6 [(1,5),(2,4),(8,6),(3,7)] => 5 [(1,5),(2,4),(8,7),(3,6)] => 4 [(1,5),(2,6),(8,3),(4,7)] => 7 [(1,5),(2,6),(8,4),(3,7)] => 6 [(1,5),(2,6),(8,7),(3,4)] => 5 [(1,5),(2,7),(8,3),(4,6)] => 8 [(1,5),(2,7),(8,4),(3,6)] => 7 [(1,5),(2,7),(8,6),(3,4)] => 6 [(1,6),(2,3),(8,4),(5,7)] => 6 [(1,6),(2,3),(8,5),(4,7)] => 5 [(1,6),(2,3),(8,7),(4,5)] => 4 [(1,6),(2,4),(8,3),(5,7)] => 7 [(1,6),(2,4),(8,5),(3,7)] => 6 [(1,6),(2,4),(8,7),(3,5)] => 5 [(1,6),(2,5),(8,3),(4,7)] => 8 [(1,6),(2,5),(8,4),(3,7)] => 7 [(1,6),(2,5),(8,7),(3,4)] => 6 [(1,6),(2,7),(8,3),(4,5)] => 9 [(1,6),(2,7),(8,4),(3,5)] => 8 [(1,6),(2,7),(8,5),(3,4)] => 7 [(1,7),(2,3),(8,4),(5,6)] => 7 [(1,7),(2,3),(8,5),(4,6)] => 6 [(1,7),(2,3),(8,6),(4,5)] => 5 [(1,7),(2,4),(8,3),(5,6)] => 8 [(1,7),(2,4),(8,5),(3,6)] => 7 [(1,7),(2,4),(8,6),(3,5)] => 6 [(1,7),(2,5),(8,3),(4,6)] => 9 [(1,7),(2,5),(8,4),(3,6)] => 8 [(1,7),(2,5),(8,6),(3,4)] => 7 [(1,7),(2,6),(8,3),(4,5)] => 10 [(1,7),(2,6),(8,4),(3,5)] => 9 [(1,7),(2,6),(8,5),(3,4)] => 8 ----------------------------------------------------------------------------- Created: Mar 01, 2013 at 03:07 by Alejandro Morales ----------------------------------------------------------------------------- Last Updated: May 29, 2015 at 16:21 by Martin Rubey