***************************************************************************** * 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: St001959 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The product of the heights of the peaks of a Dyck path. ----------------------------------------------------------------------------- References: [1] [mathoverflow:480173] [2] Sum over all Dyck paths of semilength n of products over all peaks p of y_p, where y_p is the y-coordinate of peak p. [[OEIS:A258173]] ----------------------------------------------------------------------------- Code: def statistic(D): return prod(D.heights()[i+1] for i in D.peaks()) ----------------------------------------------------------------------------- Statistic values: [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] => 4 [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] => 4 [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] => 4 [1,1,0,1,0,0,1,0] => 4 [1,1,0,1,0,1,0,0] => 8 [1,1,0,1,1,0,0,0] => 6 [1,1,1,0,0,0,1,0] => 3 [1,1,1,0,0,1,0,0] => 6 [1,1,1,0,1,0,0,0] => 9 [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] => 4 [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] => 4 [1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,1,0,0] => 8 [1,0,1,1,0,1,1,0,0,0] => 6 [1,0,1,1,1,0,0,0,1,0] => 3 [1,0,1,1,1,0,0,1,0,0] => 6 [1,0,1,1,1,0,1,0,0,0] => 9 [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] => 4 [1,1,0,0,1,1,0,0,1,0] => 4 [1,1,0,0,1,1,0,1,0,0] => 8 [1,1,0,0,1,1,1,0,0,0] => 6 [1,1,0,1,0,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,1,0,0] => 8 [1,1,0,1,0,1,0,0,1,0] => 8 [1,1,0,1,0,1,0,1,0,0] => 16 [1,1,0,1,0,1,1,0,0,0] => 12 [1,1,0,1,1,0,0,0,1,0] => 6 [1,1,0,1,1,0,0,1,0,0] => 12 [1,1,0,1,1,0,1,0,0,0] => 18 [1,1,0,1,1,1,0,0,0,0] => 8 [1,1,1,0,0,0,1,0,1,0] => 3 [1,1,1,0,0,0,1,1,0,0] => 6 [1,1,1,0,0,1,0,0,1,0] => 6 [1,1,1,0,0,1,0,1,0,0] => 12 [1,1,1,0,0,1,1,0,0,0] => 9 [1,1,1,0,1,0,0,0,1,0] => 9 [1,1,1,0,1,0,0,1,0,0] => 18 [1,1,1,0,1,0,1,0,0,0] => 27 [1,1,1,0,1,1,0,0,0,0] => 12 [1,1,1,1,0,0,0,0,1,0] => 4 [1,1,1,1,0,0,0,1,0,0] => 8 [1,1,1,1,0,0,1,0,0,0] => 12 [1,1,1,1,0,1,0,0,0,0] => 16 [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] => 4 [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] => 4 [1,0,1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,0,1,1,0,1,0,1,0,0] => 8 [1,0,1,0,1,1,0,1,1,0,0,0] => 6 [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] => 6 [1,0,1,0,1,1,1,0,1,0,0,0] => 9 [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] => 4 [1,0,1,1,0,0,1,1,0,0,1,0] => 4 [1,0,1,1,0,0,1,1,0,1,0,0] => 8 [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] => 4 [1,0,1,1,0,1,0,0,1,1,0,0] => 8 [1,0,1,1,0,1,0,1,0,0,1,0] => 8 [1,0,1,1,0,1,0,1,0,1,0,0] => 16 [1,0,1,1,0,1,0,1,1,0,0,0] => 12 [1,0,1,1,0,1,1,0,0,0,1,0] => 6 [1,0,1,1,0,1,1,0,0,1,0,0] => 12 [1,0,1,1,0,1,1,0,1,0,0,0] => 18 [1,0,1,1,0,1,1,1,0,0,0,0] => 8 [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] => 6 [1,0,1,1,1,0,0,1,0,0,1,0] => 6 [1,0,1,1,1,0,0,1,0,1,0,0] => 12 [1,0,1,1,1,0,0,1,1,0,0,0] => 9 [1,0,1,1,1,0,1,0,0,0,1,0] => 9 [1,0,1,1,1,0,1,0,0,1,0,0] => 18 [1,0,1,1,1,0,1,0,1,0,0,0] => 27 [1,0,1,1,1,0,1,1,0,0,0,0] => 12 [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] => 8 [1,0,1,1,1,1,0,0,1,0,0,0] => 12 [1,0,1,1,1,1,0,1,0,0,0,0] => 16 [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] => 4 [1,1,0,0,1,0,1,1,0,0,1,0] => 4 [1,1,0,0,1,0,1,1,0,1,0,0] => 8 [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] => 4 [1,1,0,0,1,1,0,0,1,1,0,0] => 8 [1,1,0,0,1,1,0,1,0,0,1,0] => 8 [1,1,0,0,1,1,0,1,0,1,0,0] => 16 [1,1,0,0,1,1,0,1,1,0,0,0] => 12 [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] => 12 [1,1,0,0,1,1,1,0,1,0,0,0] => 18 [1,1,0,0,1,1,1,1,0,0,0,0] => 8 [1,1,0,1,0,0,1,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,0,1,1,0,0] => 8 [1,1,0,1,0,0,1,1,0,0,1,0] => 8 [1,1,0,1,0,0,1,1,0,1,0,0] => 16 [1,1,0,1,0,0,1,1,1,0,0,0] => 12 [1,1,0,1,0,1,0,0,1,0,1,0] => 8 [1,1,0,1,0,1,0,0,1,1,0,0] => 16 [1,1,0,1,0,1,0,1,0,0,1,0] => 16 [1,1,0,1,0,1,0,1,0,1,0,0] => 32 [1,1,0,1,0,1,0,1,1,0,0,0] => 24 [1,1,0,1,0,1,1,0,0,0,1,0] => 12 [1,1,0,1,0,1,1,0,0,1,0,0] => 24 [1,1,0,1,0,1,1,0,1,0,0,0] => 36 [1,1,0,1,0,1,1,1,0,0,0,0] => 16 [1,1,0,1,1,0,0,0,1,0,1,0] => 6 [1,1,0,1,1,0,0,0,1,1,0,0] => 12 [1,1,0,1,1,0,0,1,0,0,1,0] => 12 [1,1,0,1,1,0,0,1,0,1,0,0] => 24 [1,1,0,1,1,0,0,1,1,0,0,0] => 18 [1,1,0,1,1,0,1,0,0,0,1,0] => 18 [1,1,0,1,1,0,1,0,0,1,0,0] => 36 [1,1,0,1,1,0,1,0,1,0,0,0] => 54 [1,1,0,1,1,0,1,1,0,0,0,0] => 24 [1,1,0,1,1,1,0,0,0,0,1,0] => 8 [1,1,0,1,1,1,0,0,0,1,0,0] => 16 [1,1,0,1,1,1,0,0,1,0,0,0] => 24 [1,1,0,1,1,1,0,1,0,0,0,0] => 32 [1,1,0,1,1,1,1,0,0,0,0,0] => 10 [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] => 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] => 12 [1,1,1,0,0,0,1,1,1,0,0,0] => 9 [1,1,1,0,0,1,0,0,1,0,1,0] => 6 [1,1,1,0,0,1,0,0,1,1,0,0] => 12 [1,1,1,0,0,1,0,1,0,0,1,0] => 12 [1,1,1,0,0,1,0,1,0,1,0,0] => 24 [1,1,1,0,0,1,0,1,1,0,0,0] => 18 [1,1,1,0,0,1,1,0,0,0,1,0] => 9 [1,1,1,0,0,1,1,0,0,1,0,0] => 18 [1,1,1,0,0,1,1,0,1,0,0,0] => 27 [1,1,1,0,0,1,1,1,0,0,0,0] => 12 [1,1,1,0,1,0,0,0,1,0,1,0] => 9 [1,1,1,0,1,0,0,0,1,1,0,0] => 18 [1,1,1,0,1,0,0,1,0,0,1,0] => 18 [1,1,1,0,1,0,0,1,0,1,0,0] => 36 [1,1,1,0,1,0,0,1,1,0,0,0] => 27 [1,1,1,0,1,0,1,0,0,0,1,0] => 27 [1,1,1,0,1,0,1,0,0,1,0,0] => 54 [1,1,1,0,1,0,1,0,1,0,0,0] => 81 [1,1,1,0,1,0,1,1,0,0,0,0] => 36 [1,1,1,0,1,1,0,0,0,0,1,0] => 12 [1,1,1,0,1,1,0,0,0,1,0,0] => 24 [1,1,1,0,1,1,0,0,1,0,0,0] => 36 [1,1,1,0,1,1,0,1,0,0,0,0] => 48 [1,1,1,0,1,1,1,0,0,0,0,0] => 15 [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] => 8 [1,1,1,1,0,0,0,1,0,0,1,0] => 8 [1,1,1,1,0,0,0,1,0,1,0,0] => 16 [1,1,1,1,0,0,0,1,1,0,0,0] => 12 [1,1,1,1,0,0,1,0,0,0,1,0] => 12 [1,1,1,1,0,0,1,0,0,1,0,0] => 24 [1,1,1,1,0,0,1,0,1,0,0,0] => 36 [1,1,1,1,0,0,1,1,0,0,0,0] => 16 [1,1,1,1,0,1,0,0,0,0,1,0] => 16 [1,1,1,1,0,1,0,0,0,1,0,0] => 32 [1,1,1,1,0,1,0,0,1,0,0,0] => 48 [1,1,1,1,0,1,0,1,0,0,0,0] => 64 [1,1,1,1,0,1,1,0,0,0,0,0] => 20 [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] => 10 [1,1,1,1,1,0,0,0,1,0,0,0] => 15 [1,1,1,1,1,0,0,1,0,0,0,0] => 20 [1,1,1,1,1,0,1,0,0,0,0,0] => 25 [1,1,1,1,1,1,0,0,0,0,0,0] => 6 ----------------------------------------------------------------------------- Created: Oct 07, 2024 at 09:04 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 07, 2024 at 09:04 by Martin Rubey