***************************************************************************** * 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: St000443 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels. ----------------------------------------------------------------------------- References: [1] Elizalde, S. Fixed points and excedances in restricted permutations [[MathSciNet:2880679]] [[arXiv:math/0212221]] ----------------------------------------------------------------------------- Code: def merge_one(tns): tns = sorted(tns) for j in range(len(tns)): for i in range(j): t1 = tns[i] t2 = tns[j] if t1[1] == t2[0]: tns.pop(j) tns.pop(i) tns.append((t1[0],t2[1])) return tns return tns def long_tunnels(D): tns = list(D.tunnels()) n = len(tns) + 1 while n > len(tns): n -= 1 tns = merge_one(tns) return tns def statistic(D): return len(long_tunnels(D)) ----------------------------------------------------------------------------- Statistic values: [1,0] => 1 [1,0,1,0] => 1 [1,1,0,0] => 2 [1,0,1,0,1,0] => 1 [1,0,1,1,0,0] => 2 [1,1,0,0,1,0] => 2 [1,1,0,1,0,0] => 2 [1,1,1,0,0,0] => 3 [1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,1,0,0] => 2 [1,0,1,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,0] => 2 [1,0,1,1,1,0,0,0] => 3 [1,1,0,0,1,0,1,0] => 2 [1,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,0,1,0] => 2 [1,1,0,1,0,1,0,0] => 2 [1,1,0,1,1,0,0,0] => 3 [1,1,1,0,0,0,1,0] => 3 [1,1,1,0,0,1,0,0] => 3 [1,1,1,0,1,0,0,0] => 3 [1,1,1,1,0,0,0,0] => 4 [1,0,1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,0,1,1,0,0] => 2 [1,0,1,0,1,1,0,0,1,0] => 2 [1,0,1,0,1,1,0,1,0,0] => 2 [1,0,1,0,1,1,1,0,0,0] => 3 [1,0,1,1,0,0,1,0,1,0] => 2 [1,0,1,1,0,0,1,1,0,0] => 3 [1,0,1,1,0,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,1,0,0] => 2 [1,0,1,1,0,1,1,0,0,0] => 3 [1,0,1,1,1,0,0,0,1,0] => 3 [1,0,1,1,1,0,0,1,0,0] => 3 [1,0,1,1,1,0,1,0,0,0] => 3 [1,0,1,1,1,1,0,0,0,0] => 4 [1,1,0,0,1,0,1,0,1,0] => 2 [1,1,0,0,1,0,1,1,0,0] => 3 [1,1,0,0,1,1,0,0,1,0] => 3 [1,1,0,0,1,1,0,1,0,0] => 3 [1,1,0,0,1,1,1,0,0,0] => 4 [1,1,0,1,0,0,1,0,1,0] => 2 [1,1,0,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,1,0,0,1,0] => 2 [1,1,0,1,0,1,0,1,0,0] => 2 [1,1,0,1,0,1,1,0,0,0] => 3 [1,1,0,1,1,0,0,0,1,0] => 3 [1,1,0,1,1,0,0,1,0,0] => 3 [1,1,0,1,1,0,1,0,0,0] => 3 [1,1,0,1,1,1,0,0,0,0] => 4 [1,1,1,0,0,0,1,0,1,0] => 3 [1,1,1,0,0,0,1,1,0,0] => 4 [1,1,1,0,0,1,0,0,1,0] => 3 [1,1,1,0,0,1,0,1,0,0] => 3 [1,1,1,0,0,1,1,0,0,0] => 4 [1,1,1,0,1,0,0,0,1,0] => 3 [1,1,1,0,1,0,0,1,0,0] => 3 [1,1,1,0,1,0,1,0,0,0] => 3 [1,1,1,0,1,1,0,0,0,0] => 4 [1,1,1,1,0,0,0,0,1,0] => 4 [1,1,1,1,0,0,0,1,0,0] => 4 [1,1,1,1,0,0,1,0,0,0] => 4 [1,1,1,1,0,1,0,0,0,0] => 4 [1,1,1,1,1,0,0,0,0,0] => 5 [1,0,1,0,1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,0,1,0,1,1,0,0] => 2 [1,0,1,0,1,0,1,1,0,0,1,0] => 2 [1,0,1,0,1,0,1,1,0,1,0,0] => 2 [1,0,1,0,1,0,1,1,1,0,0,0] => 3 [1,0,1,0,1,1,0,0,1,0,1,0] => 2 [1,0,1,0,1,1,0,0,1,1,0,0] => 3 [1,0,1,0,1,1,0,1,0,0,1,0] => 2 [1,0,1,0,1,1,0,1,0,1,0,0] => 2 [1,0,1,0,1,1,0,1,1,0,0,0] => 3 [1,0,1,0,1,1,1,0,0,0,1,0] => 3 [1,0,1,0,1,1,1,0,0,1,0,0] => 3 [1,0,1,0,1,1,1,0,1,0,0,0] => 3 [1,0,1,0,1,1,1,1,0,0,0,0] => 4 [1,0,1,1,0,0,1,0,1,0,1,0] => 2 [1,0,1,1,0,0,1,0,1,1,0,0] => 3 [1,0,1,1,0,0,1,1,0,0,1,0] => 3 [1,0,1,1,0,0,1,1,0,1,0,0] => 3 [1,0,1,1,0,0,1,1,1,0,0,0] => 4 [1,0,1,1,0,1,0,0,1,0,1,0] => 2 [1,0,1,1,0,1,0,0,1,1,0,0] => 3 [1,0,1,1,0,1,0,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,1,0,1,0,0] => 2 [1,0,1,1,0,1,0,1,1,0,0,0] => 3 [1,0,1,1,0,1,1,0,0,0,1,0] => 3 [1,0,1,1,0,1,1,0,0,1,0,0] => 3 [1,0,1,1,0,1,1,0,1,0,0,0] => 3 [1,0,1,1,0,1,1,1,0,0,0,0] => 4 [1,0,1,1,1,0,0,0,1,0,1,0] => 3 [1,0,1,1,1,0,0,0,1,1,0,0] => 4 [1,0,1,1,1,0,0,1,0,0,1,0] => 3 [1,0,1,1,1,0,0,1,0,1,0,0] => 3 [1,0,1,1,1,0,0,1,1,0,0,0] => 4 [1,0,1,1,1,0,1,0,0,0,1,0] => 3 [1,0,1,1,1,0,1,0,0,1,0,0] => 3 [1,0,1,1,1,0,1,0,1,0,0,0] => 3 [1,0,1,1,1,0,1,1,0,0,0,0] => 4 [1,0,1,1,1,1,0,0,0,0,1,0] => 4 [1,0,1,1,1,1,0,0,0,1,0,0] => 4 [1,0,1,1,1,1,0,0,1,0,0,0] => 4 [1,0,1,1,1,1,0,1,0,0,0,0] => 4 [1,0,1,1,1,1,1,0,0,0,0,0] => 5 [1,1,0,0,1,0,1,0,1,0,1,0] => 2 [1,1,0,0,1,0,1,0,1,1,0,0] => 3 [1,1,0,0,1,0,1,1,0,0,1,0] => 3 [1,1,0,0,1,0,1,1,0,1,0,0] => 3 [1,1,0,0,1,0,1,1,1,0,0,0] => 4 [1,1,0,0,1,1,0,0,1,0,1,0] => 3 [1,1,0,0,1,1,0,0,1,1,0,0] => 4 [1,1,0,0,1,1,0,1,0,0,1,0] => 3 [1,1,0,0,1,1,0,1,0,1,0,0] => 3 [1,1,0,0,1,1,0,1,1,0,0,0] => 4 [1,1,0,0,1,1,1,0,0,0,1,0] => 4 [1,1,0,0,1,1,1,0,0,1,0,0] => 4 [1,1,0,0,1,1,1,0,1,0,0,0] => 4 [1,1,0,0,1,1,1,1,0,0,0,0] => 5 [1,1,0,1,0,0,1,0,1,0,1,0] => 2 [1,1,0,1,0,0,1,0,1,1,0,0] => 3 [1,1,0,1,0,0,1,1,0,0,1,0] => 3 [1,1,0,1,0,0,1,1,0,1,0,0] => 3 [1,1,0,1,0,0,1,1,1,0,0,0] => 4 [1,1,0,1,0,1,0,0,1,0,1,0] => 2 [1,1,0,1,0,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,1,0,1,0,0,1,0] => 2 [1,1,0,1,0,1,0,1,0,1,0,0] => 2 [1,1,0,1,0,1,0,1,1,0,0,0] => 3 [1,1,0,1,0,1,1,0,0,0,1,0] => 3 [1,1,0,1,0,1,1,0,0,1,0,0] => 3 [1,1,0,1,0,1,1,0,1,0,0,0] => 3 [1,1,0,1,0,1,1,1,0,0,0,0] => 4 [1,1,0,1,1,0,0,0,1,0,1,0] => 3 [1,1,0,1,1,0,0,0,1,1,0,0] => 4 [1,1,0,1,1,0,0,1,0,0,1,0] => 3 [1,1,0,1,1,0,0,1,0,1,0,0] => 3 [1,1,0,1,1,0,0,1,1,0,0,0] => 4 [1,1,0,1,1,0,1,0,0,0,1,0] => 3 [1,1,0,1,1,0,1,0,0,1,0,0] => 3 [1,1,0,1,1,0,1,0,1,0,0,0] => 3 [1,1,0,1,1,0,1,1,0,0,0,0] => 4 [1,1,0,1,1,1,0,0,0,0,1,0] => 4 [1,1,0,1,1,1,0,0,0,1,0,0] => 4 [1,1,0,1,1,1,0,0,1,0,0,0] => 4 [1,1,0,1,1,1,0,1,0,0,0,0] => 4 [1,1,0,1,1,1,1,0,0,0,0,0] => 5 [1,1,1,0,0,0,1,0,1,0,1,0] => 3 [1,1,1,0,0,0,1,0,1,1,0,0] => 4 [1,1,1,0,0,0,1,1,0,0,1,0] => 4 [1,1,1,0,0,0,1,1,0,1,0,0] => 4 [1,1,1,0,0,0,1,1,1,0,0,0] => 5 [1,1,1,0,0,1,0,0,1,0,1,0] => 3 [1,1,1,0,0,1,0,0,1,1,0,0] => 4 [1,1,1,0,0,1,0,1,0,0,1,0] => 3 [1,1,1,0,0,1,0,1,0,1,0,0] => 3 [1,1,1,0,0,1,0,1,1,0,0,0] => 4 [1,1,1,0,0,1,1,0,0,0,1,0] => 4 [1,1,1,0,0,1,1,0,0,1,0,0] => 4 [1,1,1,0,0,1,1,0,1,0,0,0] => 4 [1,1,1,0,0,1,1,1,0,0,0,0] => 5 [1,1,1,0,1,0,0,0,1,0,1,0] => 3 [1,1,1,0,1,0,0,0,1,1,0,0] => 4 [1,1,1,0,1,0,0,1,0,0,1,0] => 3 [1,1,1,0,1,0,0,1,0,1,0,0] => 3 [1,1,1,0,1,0,0,1,1,0,0,0] => 4 [1,1,1,0,1,0,1,0,0,0,1,0] => 3 [1,1,1,0,1,0,1,0,0,1,0,0] => 3 [1,1,1,0,1,0,1,0,1,0,0,0] => 3 [1,1,1,0,1,0,1,1,0,0,0,0] => 4 [1,1,1,0,1,1,0,0,0,0,1,0] => 4 [1,1,1,0,1,1,0,0,0,1,0,0] => 4 [1,1,1,0,1,1,0,0,1,0,0,0] => 4 [1,1,1,0,1,1,0,1,0,0,0,0] => 4 [1,1,1,0,1,1,1,0,0,0,0,0] => 5 [1,1,1,1,0,0,0,0,1,0,1,0] => 4 [1,1,1,1,0,0,0,0,1,1,0,0] => 5 [1,1,1,1,0,0,0,1,0,0,1,0] => 4 [1,1,1,1,0,0,0,1,0,1,0,0] => 4 [1,1,1,1,0,0,0,1,1,0,0,0] => 5 [1,1,1,1,0,0,1,0,0,0,1,0] => 4 [1,1,1,1,0,0,1,0,0,1,0,0] => 4 [1,1,1,1,0,0,1,0,1,0,0,0] => 4 [1,1,1,1,0,0,1,1,0,0,0,0] => 5 [1,1,1,1,0,1,0,0,0,0,1,0] => 4 [1,1,1,1,0,1,0,0,0,1,0,0] => 4 [1,1,1,1,0,1,0,0,1,0,0,0] => 4 [1,1,1,1,0,1,0,1,0,0,0,0] => 4 [1,1,1,1,0,1,1,0,0,0,0,0] => 5 [1,1,1,1,1,0,0,0,0,0,1,0] => 5 [1,1,1,1,1,0,0,0,0,1,0,0] => 5 [1,1,1,1,1,0,0,0,1,0,0,0] => 5 [1,1,1,1,1,0,0,1,0,0,0,0] => 5 [1,1,1,1,1,0,1,0,0,0,0,0] => 5 [1,1,1,1,1,1,0,0,0,0,0,0] => 6 ----------------------------------------------------------------------------- Created: Mar 07, 2016 at 20:02 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jan 13, 2021 at 18:12 by Martin Rubey