***************************************************************************** * 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: St000144 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form \$1^h0^h\$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics. ----------------------------------------------------------------------------- References: [1] Denise, A., Simion, R. Two combinatorial statistics on Dyck paths [[MathSciNet:1312450]] [2] Le Borgne, Y. Counting upper interactions in Dyck paths [[MathSciNet:2196524]] ----------------------------------------------------------------------------- Code: def statistic(x): return x.pyramid_weight() ----------------------------------------------------------------------------- Statistic values: [1,0] => 1 [1,0,1,0] => 2 [1,1,0,0] => 2 [1,0,1,0,1,0] => 3 [1,0,1,1,0,0] => 3 [1,1,0,0,1,0] => 3 [1,1,0,1,0,0] => 2 [1,1,1,0,0,0] => 3 [1,0,1,0,1,0,1,0] => 4 [1,0,1,0,1,1,0,0] => 4 [1,0,1,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,0] => 3 [1,0,1,1,1,0,0,0] => 4 [1,1,0,0,1,0,1,0] => 4 [1,1,0,0,1,1,0,0] => 4 [1,1,0,1,0,0,1,0] => 3 [1,1,0,1,0,1,0,0] => 3 [1,1,0,1,1,0,0,0] => 3 [1,1,1,0,0,0,1,0] => 4 [1,1,1,0,0,1,0,0] => 3 [1,1,1,0,1,0,0,0] => 2 [1,1,1,1,0,0,0,0] => 4 [1,0,1,0,1,0,1,0,1,0] => 5 [1,0,1,0,1,0,1,1,0,0] => 5 [1,0,1,0,1,1,0,0,1,0] => 5 [1,0,1,0,1,1,0,1,0,0] => 4 [1,0,1,0,1,1,1,0,0,0] => 5 [1,0,1,1,0,0,1,0,1,0] => 5 [1,0,1,1,0,0,1,1,0,0] => 5 [1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,1,0,0] => 4 [1,0,1,1,0,1,1,0,0,0] => 4 [1,0,1,1,1,0,0,0,1,0] => 5 [1,0,1,1,1,0,0,1,0,0] => 4 [1,0,1,1,1,0,1,0,0,0] => 3 [1,0,1,1,1,1,0,0,0,0] => 5 [1,1,0,0,1,0,1,0,1,0] => 5 [1,1,0,0,1,0,1,1,0,0] => 5 [1,1,0,0,1,1,0,0,1,0] => 5 [1,1,0,0,1,1,0,1,0,0] => 4 [1,1,0,0,1,1,1,0,0,0] => 5 [1,1,0,1,0,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,1,0,0] => 4 [1,1,0,1,0,1,0,0,1,0] => 4 [1,1,0,1,0,1,0,1,0,0] => 4 [1,1,0,1,0,1,1,0,0,0] => 4 [1,1,0,1,1,0,0,0,1,0] => 4 [1,1,0,1,1,0,0,1,0,0] => 4 [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] => 5 [1,1,1,0,0,0,1,1,0,0] => 5 [1,1,1,0,0,1,0,0,1,0] => 4 [1,1,1,0,0,1,0,1,0,0] => 4 [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] => 3 [1,1,1,1,0,0,0,0,1,0] => 5 [1,1,1,1,0,0,0,1,0,0] => 4 [1,1,1,1,0,0,1,0,0,0] => 3 [1,1,1,1,0,1,0,0,0,0] => 2 [1,1,1,1,1,0,0,0,0,0] => 5 [1,0,1,0,1,0,1,0,1,0,1,0] => 6 [1,0,1,0,1,0,1,0,1,1,0,0] => 6 [1,0,1,0,1,0,1,1,0,0,1,0] => 6 [1,0,1,0,1,0,1,1,0,1,0,0] => 5 [1,0,1,0,1,0,1,1,1,0,0,0] => 6 [1,0,1,0,1,1,0,0,1,0,1,0] => 6 [1,0,1,0,1,1,0,0,1,1,0,0] => 6 [1,0,1,0,1,1,0,1,0,0,1,0] => 5 [1,0,1,0,1,1,0,1,0,1,0,0] => 5 [1,0,1,0,1,1,0,1,1,0,0,0] => 5 [1,0,1,0,1,1,1,0,0,0,1,0] => 6 [1,0,1,0,1,1,1,0,0,1,0,0] => 5 [1,0,1,0,1,1,1,0,1,0,0,0] => 4 [1,0,1,0,1,1,1,1,0,0,0,0] => 6 [1,0,1,1,0,0,1,0,1,0,1,0] => 6 [1,0,1,1,0,0,1,0,1,1,0,0] => 6 [1,0,1,1,0,0,1,1,0,0,1,0] => 6 [1,0,1,1,0,0,1,1,0,1,0,0] => 5 [1,0,1,1,0,0,1,1,1,0,0,0] => 6 [1,0,1,1,0,1,0,0,1,0,1,0] => 5 [1,0,1,1,0,1,0,0,1,1,0,0] => 5 [1,0,1,1,0,1,0,1,0,0,1,0] => 5 [1,0,1,1,0,1,0,1,0,1,0,0] => 5 [1,0,1,1,0,1,0,1,1,0,0,0] => 5 [1,0,1,1,0,1,1,0,0,0,1,0] => 5 [1,0,1,1,0,1,1,0,0,1,0,0] => 5 [1,0,1,1,0,1,1,0,1,0,0,0] => 4 [1,0,1,1,0,1,1,1,0,0,0,0] => 5 [1,0,1,1,1,0,0,0,1,0,1,0] => 6 [1,0,1,1,1,0,0,0,1,1,0,0] => 6 [1,0,1,1,1,0,0,1,0,0,1,0] => 5 [1,0,1,1,1,0,0,1,0,1,0,0] => 5 [1,0,1,1,1,0,0,1,1,0,0,0] => 5 [1,0,1,1,1,0,1,0,0,0,1,0] => 4 [1,0,1,1,1,0,1,0,0,1,0,0] => 4 [1,0,1,1,1,0,1,0,1,0,0,0] => 4 [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] => 6 [1,0,1,1,1,1,0,0,0,1,0,0] => 5 [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] => 3 [1,0,1,1,1,1,1,0,0,0,0,0] => 6 [1,1,0,0,1,0,1,0,1,0,1,0] => 6 [1,1,0,0,1,0,1,0,1,1,0,0] => 6 [1,1,0,0,1,0,1,1,0,0,1,0] => 6 [1,1,0,0,1,0,1,1,0,1,0,0] => 5 [1,1,0,0,1,0,1,1,1,0,0,0] => 6 [1,1,0,0,1,1,0,0,1,0,1,0] => 6 [1,1,0,0,1,1,0,0,1,1,0,0] => 6 [1,1,0,0,1,1,0,1,0,0,1,0] => 5 [1,1,0,0,1,1,0,1,0,1,0,0] => 5 [1,1,0,0,1,1,0,1,1,0,0,0] => 5 [1,1,0,0,1,1,1,0,0,0,1,0] => 6 [1,1,0,0,1,1,1,0,0,1,0,0] => 5 [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] => 6 [1,1,0,1,0,0,1,0,1,0,1,0] => 5 [1,1,0,1,0,0,1,0,1,1,0,0] => 5 [1,1,0,1,0,0,1,1,0,0,1,0] => 5 [1,1,0,1,0,0,1,1,0,1,0,0] => 4 [1,1,0,1,0,0,1,1,1,0,0,0] => 5 [1,1,0,1,0,1,0,0,1,0,1,0] => 5 [1,1,0,1,0,1,0,0,1,1,0,0] => 5 [1,1,0,1,0,1,0,1,0,0,1,0] => 5 [1,1,0,1,0,1,0,1,0,1,0,0] => 5 [1,1,0,1,0,1,0,1,1,0,0,0] => 5 [1,1,0,1,0,1,1,0,0,0,1,0] => 5 [1,1,0,1,0,1,1,0,0,1,0,0] => 5 [1,1,0,1,0,1,1,0,1,0,0,0] => 4 [1,1,0,1,0,1,1,1,0,0,0,0] => 5 [1,1,0,1,1,0,0,0,1,0,1,0] => 5 [1,1,0,1,1,0,0,0,1,1,0,0] => 5 [1,1,0,1,1,0,0,1,0,0,1,0] => 5 [1,1,0,1,1,0,0,1,0,1,0,0] => 5 [1,1,0,1,1,0,0,1,1,0,0,0] => 5 [1,1,0,1,1,0,1,0,0,0,1,0] => 4 [1,1,0,1,1,0,1,0,0,1,0,0] => 4 [1,1,0,1,1,0,1,0,1,0,0,0] => 4 [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] => 5 [1,1,0,1,1,1,0,0,0,1,0,0] => 5 [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] => 3 [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] => 6 [1,1,1,0,0,0,1,0,1,1,0,0] => 6 [1,1,1,0,0,0,1,1,0,0,1,0] => 6 [1,1,1,0,0,0,1,1,0,1,0,0] => 5 [1,1,1,0,0,0,1,1,1,0,0,0] => 6 [1,1,1,0,0,1,0,0,1,0,1,0] => 5 [1,1,1,0,0,1,0,0,1,1,0,0] => 5 [1,1,1,0,0,1,0,1,0,0,1,0] => 5 [1,1,1,0,0,1,0,1,0,1,0,0] => 5 [1,1,1,0,0,1,0,1,1,0,0,0] => 5 [1,1,1,0,0,1,1,0,0,0,1,0] => 5 [1,1,1,0,0,1,1,0,0,1,0,0] => 5 [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] => 4 [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] => 4 [1,1,1,0,1,0,0,1,0,1,0,0] => 4 [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] => 4 [1,1,1,0,1,0,1,0,0,1,0,0] => 4 [1,1,1,0,1,0,1,0,1,0,0,0] => 4 [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] => 3 [1,1,1,0,1,1,1,0,0,0,0,0] => 4 [1,1,1,1,0,0,0,0,1,0,1,0] => 6 [1,1,1,1,0,0,0,0,1,1,0,0] => 6 [1,1,1,1,0,0,0,1,0,0,1,0] => 5 [1,1,1,1,0,0,0,1,0,1,0,0] => 5 [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] => 4 [1,1,1,1,0,1,0,0,0,0,1,0] => 3 [1,1,1,1,0,1,0,0,0,1,0,0] => 3 [1,1,1,1,0,1,0,0,1,0,0,0] => 3 [1,1,1,1,0,1,0,1,0,0,0,0] => 3 [1,1,1,1,0,1,1,0,0,0,0,0] => 3 [1,1,1,1,1,0,0,0,0,0,1,0] => 6 [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] => 4 [1,1,1,1,1,0,0,1,0,0,0,0] => 3 [1,1,1,1,1,0,1,0,0,0,0,0] => 2 [1,1,1,1,1,1,0,0,0,0,0,0] => 6 ----------------------------------------------------------------------------- Created: Jul 01, 2013 at 12:03 by Olivier Mallet ----------------------------------------------------------------------------- Last Updated: Apr 07, 2016 at 08:33 by Martin Rubey