***************************************************************************** * 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: St000806 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps. ----------------------------------------------------------------------------- References: [1] Deutsch, E., Elizalde, S. A bijection between bargraphs and Dyck paths [[arXiv:1705.05984]] ----------------------------------------------------------------------------- Code: def statistic(c): if len(c) == 0: return 0 sp = 1+c[0] for i in range(1, len(c)): if c[i] > c[i-1]: sp += 1+c[i]-c[i-1] else: sp += 1 return sp ----------------------------------------------------------------------------- Statistic values: [1,1] => 3 [2] => 3 [1,1,1] => 4 [1,2] => 4 [2,1] => 4 [3] => 4 [1,1,1,1] => 5 [1,1,2] => 5 [1,2,1] => 5 [1,3] => 5 [2,1,1] => 5 [2,2] => 4 [3,1] => 5 [4] => 5 [1,1,1,1,1] => 6 [1,1,1,2] => 6 [1,1,2,1] => 6 [1,1,3] => 6 [1,2,1,1] => 6 [1,2,2] => 5 [1,3,1] => 6 [1,4] => 6 [2,1,1,1] => 6 [2,1,2] => 6 [2,2,1] => 5 [2,3] => 5 [3,1,1] => 6 [3,2] => 5 [4,1] => 6 [5] => 6 [1,1,1,1,1,1] => 7 [1,1,1,1,2] => 7 [1,1,1,2,1] => 7 [1,1,1,3] => 7 [1,1,2,1,1] => 7 [1,1,2,2] => 6 [1,1,3,1] => 7 [1,1,4] => 7 [1,2,1,1,1] => 7 [1,2,1,2] => 7 [1,2,2,1] => 6 [1,2,3] => 6 [1,3,1,1] => 7 [1,3,2] => 6 [1,4,1] => 7 [1,5] => 7 [2,1,1,1,1] => 7 [2,1,1,2] => 7 [2,1,2,1] => 7 [2,1,3] => 7 [2,2,1,1] => 6 [2,2,2] => 5 [2,3,1] => 6 [2,4] => 6 [3,1,1,1] => 7 [3,1,2] => 7 [3,2,1] => 6 [3,3] => 5 [4,1,1] => 7 [4,2] => 6 [5,1] => 7 [6] => 7 [1,1,1,1,1,1,1] => 8 [1,1,1,1,1,2] => 8 [1,1,1,1,2,1] => 8 [1,1,1,1,3] => 8 [1,1,1,2,1,1] => 8 [1,1,1,2,2] => 7 [1,1,1,3,1] => 8 [1,1,1,4] => 8 [1,1,2,1,1,1] => 8 [1,1,2,1,2] => 8 [1,1,2,2,1] => 7 [1,1,2,3] => 7 [1,1,3,1,1] => 8 [1,1,3,2] => 7 [1,1,4,1] => 8 [1,1,5] => 8 [1,2,1,1,1,1] => 8 [1,2,1,1,2] => 8 [1,2,1,2,1] => 8 [1,2,1,3] => 8 [1,2,2,1,1] => 7 [1,2,2,2] => 6 [1,2,3,1] => 7 [1,2,4] => 7 [1,3,1,1,1] => 8 [1,3,1,2] => 8 [1,3,2,1] => 7 [1,3,3] => 6 [1,4,1,1] => 8 [1,4,2] => 7 [1,5,1] => 8 [1,6] => 8 [2,1,1,1,1,1] => 8 [2,1,1,1,2] => 8 [2,1,1,2,1] => 8 [2,1,1,3] => 8 [2,1,2,1,1] => 8 [2,1,2,2] => 7 [2,1,3,1] => 8 [2,1,4] => 8 [2,2,1,1,1] => 7 [2,2,1,2] => 7 [2,2,2,1] => 6 [2,2,3] => 6 [2,3,1,1] => 7 [2,3,2] => 6 [2,4,1] => 7 [2,5] => 7 [3,1,1,1,1] => 8 [3,1,1,2] => 8 [3,1,2,1] => 8 [3,1,3] => 8 [3,2,1,1] => 7 [3,2,2] => 6 [3,3,1] => 6 [3,4] => 6 [4,1,1,1] => 8 [4,1,2] => 8 [4,2,1] => 7 [4,3] => 6 [5,1,1] => 8 [5,2] => 7 [6,1] => 8 [7] => 8 [1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,2] => 9 [1,1,1,1,1,2,1] => 9 [1,1,1,1,1,3] => 9 [1,1,1,1,2,1,1] => 9 [1,1,1,1,2,2] => 8 [1,1,1,1,3,1] => 9 [1,1,1,1,4] => 9 [1,1,1,2,1,1,1] => 9 [1,1,1,2,1,2] => 9 [1,1,1,2,2,1] => 8 [1,1,1,2,3] => 8 [1,1,1,3,1,1] => 9 [1,1,1,3,2] => 8 [1,1,1,4,1] => 9 [1,1,1,5] => 9 [1,1,2,1,1,1,1] => 9 [1,1,2,1,1,2] => 9 [1,1,2,1,2,1] => 9 [1,1,2,1,3] => 9 [1,1,2,2,1,1] => 8 [1,1,2,2,2] => 7 [1,1,2,3,1] => 8 [1,1,2,4] => 8 [1,1,3,1,1,1] => 9 [1,1,3,1,2] => 9 [1,1,3,2,1] => 8 [1,1,3,3] => 7 [1,1,4,1,1] => 9 [1,1,4,2] => 8 [1,1,5,1] => 9 [1,1,6] => 9 [1,2,1,1,1,1,1] => 9 [1,2,1,1,1,2] => 9 [1,2,1,1,2,1] => 9 [1,2,1,1,3] => 9 [1,2,1,2,1,1] => 9 [1,2,1,2,2] => 8 [1,2,1,3,1] => 9 [1,2,1,4] => 9 [1,2,2,1,1,1] => 8 [1,2,2,1,2] => 8 [1,2,2,2,1] => 7 [1,2,2,3] => 7 [1,2,3,1,1] => 8 [1,2,3,2] => 7 [1,2,4,1] => 8 [1,2,5] => 8 [1,3,1,1,1,1] => 9 [1,3,1,1,2] => 9 [1,3,1,2,1] => 9 [1,3,1,3] => 9 [1,3,2,1,1] => 8 [1,3,2,2] => 7 [1,3,3,1] => 7 [1,3,4] => 7 [1,4,1,1,1] => 9 [1,4,1,2] => 9 [1,4,2,1] => 8 [1,4,3] => 7 [1,5,1,1] => 9 [1,5,2] => 8 [1,6,1] => 9 [1,7] => 9 [2,1,1,1,1,1,1] => 9 [2,1,1,1,1,2] => 9 [2,1,1,1,2,1] => 9 [2,1,1,1,3] => 9 [2,1,1,2,1,1] => 9 [2,1,1,2,2] => 8 [2,1,1,3,1] => 9 [2,1,1,4] => 9 [2,1,2,1,1,1] => 9 [2,1,2,1,2] => 9 [2,1,2,2,1] => 8 [2,1,2,3] => 8 [2,1,3,1,1] => 9 [2,1,3,2] => 8 [2,1,4,1] => 9 [2,1,5] => 9 [2,2,1,1,1,1] => 8 [2,2,1,1,2] => 8 [2,2,1,2,1] => 8 [2,2,1,3] => 8 [2,2,2,1,1] => 7 [2,2,2,2] => 6 [2,2,3,1] => 7 [2,2,4] => 7 [2,3,1,1,1] => 8 [2,3,1,2] => 8 [2,3,2,1] => 7 [2,3,3] => 6 [2,4,1,1] => 8 [2,4,2] => 7 [2,5,1] => 8 [2,6] => 8 [3,1,1,1,1,1] => 9 [3,1,1,1,2] => 9 [3,1,1,2,1] => 9 [3,1,1,3] => 9 [3,1,2,1,1] => 9 [3,1,2,2] => 8 [3,1,3,1] => 9 [3,1,4] => 9 [3,2,1,1,1] => 8 [3,2,1,2] => 8 [3,2,2,1] => 7 [3,2,3] => 7 [3,3,1,1] => 7 [3,3,2] => 6 [3,4,1] => 7 [3,5] => 7 [4,1,1,1,1] => 9 [4,1,1,2] => 9 [4,1,2,1] => 9 [4,1,3] => 9 [4,2,1,1] => 8 [4,2,2] => 7 [4,3,1] => 7 [4,4] => 6 [5,1,1,1] => 9 [5,1,2] => 9 [5,2,1] => 8 [5,3] => 7 [6,1,1] => 9 [6,2] => 8 [7,1] => 9 [8] => 9 [1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,2] => 10 [1,1,1,1,1,1,2,1] => 10 [1,1,1,1,1,1,3] => 10 [1,1,1,1,1,2,1,1] => 10 [1,1,1,1,1,2,2] => 9 [1,1,1,1,1,3,1] => 10 [1,1,1,1,1,4] => 10 [1,1,1,1,2,1,1,1] => 10 [1,1,1,1,2,1,2] => 10 [1,1,1,1,2,2,1] => 9 [1,1,1,1,2,3] => 9 [1,1,1,1,3,1,1] => 10 [1,1,1,1,3,2] => 9 [1,1,1,1,4,1] => 10 [1,1,1,1,5] => 10 [1,1,1,2,1,1,1,1] => 10 [1,1,1,2,1,1,2] => 10 [1,1,1,2,1,2,1] => 10 [1,1,1,2,1,3] => 10 [1,1,1,2,2,1,1] => 9 [1,1,1,2,2,2] => 8 [1,1,1,2,3,1] => 9 [1,1,1,2,4] => 9 [1,1,1,3,1,1,1] => 10 [1,1,1,3,1,2] => 10 [1,1,1,3,2,1] => 9 [1,1,1,3,3] => 8 [1,1,1,4,1,1] => 10 [1,1,1,4,2] => 9 [1,1,1,5,1] => 10 [1,1,1,6] => 10 [1,1,2,1,1,1,1,1] => 10 [1,1,2,1,1,1,2] => 10 [1,1,2,1,1,2,1] => 10 [1,1,2,1,1,3] => 10 [1,1,2,1,2,1,1] => 10 [1,1,2,1,2,2] => 9 [1,1,2,1,3,1] => 10 [1,1,2,1,4] => 10 [1,1,2,2,1,1,1] => 9 [1,1,2,2,1,2] => 9 [1,1,2,2,2,1] => 8 [1,1,2,2,3] => 8 [1,1,2,3,1,1] => 9 [1,1,2,3,2] => 8 [1,1,2,4,1] => 9 [1,1,2,5] => 9 [1,1,3,1,1,1,1] => 10 [1,1,3,1,1,2] => 10 [1,1,3,1,2,1] => 10 [1,1,3,1,3] => 10 [1,1,3,2,1,1] => 9 [1,1,3,2,2] => 8 [1,1,3,3,1] => 8 [1,1,3,4] => 8 [1,1,4,1,1,1] => 10 [1,1,4,1,2] => 10 [1,1,4,2,1] => 9 [1,1,4,3] => 8 [1,1,5,1,1] => 10 [1,1,5,2] => 9 [1,1,6,1] => 10 [1,1,7] => 10 [1,2,1,1,1,1,1,1] => 10 [1,2,1,1,1,1,2] => 10 [1,2,1,1,1,2,1] => 10 [1,2,1,1,1,3] => 10 [1,2,1,1,2,1,1] => 10 [1,2,1,1,2,2] => 9 [1,2,1,1,3,1] => 10 [1,2,1,1,4] => 10 [1,2,1,2,1,1,1] => 10 [1,2,1,2,1,2] => 10 [1,2,1,2,2,1] => 9 [1,2,1,2,3] => 9 [1,2,1,3,1,1] => 10 [1,2,1,3,2] => 9 [1,2,1,4,1] => 10 [1,2,1,5] => 10 [1,2,2,1,1,1,1] => 9 [1,2,2,1,1,2] => 9 [1,2,2,1,2,1] => 9 [1,2,2,1,3] => 9 [1,2,2,2,1,1] => 8 [1,2,2,2,2] => 7 [1,2,2,3,1] => 8 [1,2,2,4] => 8 [1,2,3,1,1,1] => 9 [1,2,3,1,2] => 9 [1,2,3,2,1] => 8 [1,2,3,3] => 7 [1,2,4,1,1] => 9 [1,2,4,2] => 8 [1,2,5,1] => 9 [1,2,6] => 9 [1,3,1,1,1,1,1] => 10 [1,3,1,1,1,2] => 10 [1,3,1,1,2,1] => 10 [1,3,1,1,3] => 10 [1,3,1,2,1,1] => 10 [1,3,1,2,2] => 9 [1,3,1,3,1] => 10 [1,3,1,4] => 10 [1,3,2,1,1,1] => 9 [1,3,2,1,2] => 9 [1,3,2,2,1] => 8 [1,3,2,3] => 8 [1,3,3,1,1] => 8 [1,3,3,2] => 7 [1,3,4,1] => 8 [1,3,5] => 8 [1,4,1,1,1,1] => 10 [1,4,1,1,2] => 10 [1,4,1,2,1] => 10 [1,4,1,3] => 10 [1,4,2,1,1] => 9 [1,4,2,2] => 8 [1,4,3,1] => 8 [1,4,4] => 7 [1,5,1,1,1] => 10 [1,5,1,2] => 10 [1,5,2,1] => 9 [1,5,3] => 8 [1,6,1,1] => 10 [1,6,2] => 9 [1,7,1] => 10 [1,8] => 10 [2,1,1,1,1,1,1,1] => 10 [2,1,1,1,1,1,2] => 10 [2,1,1,1,1,2,1] => 10 [2,1,1,1,1,3] => 10 [2,1,1,1,2,1,1] => 10 [2,1,1,1,2,2] => 9 [2,1,1,1,3,1] => 10 [2,1,1,1,4] => 10 [2,1,1,2,1,1,1] => 10 [2,1,1,2,1,2] => 10 [2,1,1,2,2,1] => 9 [2,1,1,2,3] => 9 [2,1,1,3,1,1] => 10 [2,1,1,3,2] => 9 [2,1,1,4,1] => 10 [2,1,1,5] => 10 [2,1,2,1,1,1,1] => 10 [2,1,2,1,1,2] => 10 [2,1,2,1,2,1] => 10 [2,1,2,1,3] => 10 [2,1,2,2,1,1] => 9 [2,1,2,2,2] => 8 [2,1,2,3,1] => 9 [2,1,2,4] => 9 [2,1,3,1,1,1] => 10 [2,1,3,1,2] => 10 [2,1,3,2,1] => 9 [2,1,3,3] => 8 [2,1,4,1,1] => 10 [2,1,4,2] => 9 [2,1,5,1] => 10 [2,1,6] => 10 [2,2,1,1,1,1,1] => 9 [2,2,1,1,1,2] => 9 [2,2,1,1,2,1] => 9 [2,2,1,1,3] => 9 [2,2,1,2,1,1] => 9 [2,2,1,2,2] => 8 [2,2,1,3,1] => 9 [2,2,1,4] => 9 [2,2,2,1,1,1] => 8 [2,2,2,1,2] => 8 [2,2,2,2,1] => 7 [2,2,2,3] => 7 [2,2,3,1,1] => 8 [2,2,3,2] => 7 [2,2,4,1] => 8 [2,2,5] => 8 [2,3,1,1,1,1] => 9 [2,3,1,1,2] => 9 [2,3,1,2,1] => 9 [2,3,1,3] => 9 [2,3,2,1,1] => 8 [2,3,2,2] => 7 [2,3,3,1] => 7 [2,3,4] => 7 [2,4,1,1,1] => 9 [2,4,1,2] => 9 [2,4,2,1] => 8 [2,4,3] => 7 [2,5,1,1] => 9 [2,5,2] => 8 [2,6,1] => 9 [2,7] => 9 [3,1,1,1,1,1,1] => 10 [3,1,1,1,1,2] => 10 [3,1,1,1,2,1] => 10 [3,1,1,1,3] => 10 [3,1,1,2,1,1] => 10 [3,1,1,2,2] => 9 [3,1,1,3,1] => 10 [3,1,1,4] => 10 [3,1,2,1,1,1] => 10 [3,1,2,1,2] => 10 [3,1,2,2,1] => 9 [3,1,2,3] => 9 [3,1,3,1,1] => 10 [3,1,3,2] => 9 [3,1,4,1] => 10 [3,1,5] => 10 [3,2,1,1,1,1] => 9 [3,2,1,1,2] => 9 [3,2,1,2,1] => 9 [3,2,1,3] => 9 [3,2,2,1,1] => 8 [3,2,2,2] => 7 [3,2,3,1] => 8 [3,2,4] => 8 [3,3,1,1,1] => 8 [3,3,1,2] => 8 [3,3,2,1] => 7 [3,3,3] => 6 [3,4,1,1] => 8 [3,4,2] => 7 [3,5,1] => 8 [3,6] => 8 [4,1,1,1,1,1] => 10 [4,1,1,1,2] => 10 [4,1,1,2,1] => 10 [4,1,1,3] => 10 [4,1,2,1,1] => 10 [4,1,2,2] => 9 [4,1,3,1] => 10 [4,1,4] => 10 [4,2,1,1,1] => 9 [4,2,1,2] => 9 [4,2,2,1] => 8 [4,2,3] => 8 [4,3,1,1] => 8 [4,3,2] => 7 [4,4,1] => 7 [4,5] => 7 [5,1,1,1,1] => 10 [5,1,1,2] => 10 [5,1,2,1] => 10 [5,1,3] => 10 [5,2,1,1] => 9 [5,2,2] => 8 [5,3,1] => 8 [5,4] => 7 [6,1,1,1] => 10 [6,1,2] => 10 [6,2,1] => 9 [6,3] => 8 [7,1,1] => 10 [7,2] => 9 [8,1] => 10 [9] => 10 ----------------------------------------------------------------------------- Created: May 18, 2017 at 08:32 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 22, 2017 at 07:22 by Martin Rubey