***************************************************************************** * 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: St000765 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The number of weak records in an integer composition. A weak record is an element \$a_i\$ such that \$a_i \geq a_j\$ for all \$j < i\$. ----------------------------------------------------------------------------- References: [1] Knopfmacher, A., Mansour, T. Record statistics in integer compositions [[MathSciNet:2721540]] ----------------------------------------------------------------------------- Code: def statistic(pi): pi = list(pi) i = 0 res = 0 for j in range(len(pi)): if pi[j] >= i: i = pi[j] res += 1 return res ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,1] => 2 [2] => 1 [1,1,1] => 3 [1,2] => 2 [2,1] => 1 [3] => 1 [1,1,1,1] => 4 [1,1,2] => 3 [1,2,1] => 2 [1,3] => 2 [2,1,1] => 1 [2,2] => 2 [3,1] => 1 [4] => 1 [1,1,1,1,1] => 5 [1,1,1,2] => 4 [1,1,2,1] => 3 [1,1,3] => 3 [1,2,1,1] => 2 [1,2,2] => 3 [1,3,1] => 2 [1,4] => 2 [2,1,1,1] => 1 [2,1,2] => 2 [2,2,1] => 2 [2,3] => 2 [3,1,1] => 1 [3,2] => 1 [4,1] => 1 [5] => 1 [1,1,1,1,1,1] => 6 [1,1,1,1,2] => 5 [1,1,1,2,1] => 4 [1,1,1,3] => 4 [1,1,2,1,1] => 3 [1,1,2,2] => 4 [1,1,3,1] => 3 [1,1,4] => 3 [1,2,1,1,1] => 2 [1,2,1,2] => 3 [1,2,2,1] => 3 [1,2,3] => 3 [1,3,1,1] => 2 [1,3,2] => 2 [1,4,1] => 2 [1,5] => 2 [2,1,1,1,1] => 1 [2,1,1,2] => 2 [2,1,2,1] => 2 [2,1,3] => 2 [2,2,1,1] => 2 [2,2,2] => 3 [2,3,1] => 2 [2,4] => 2 [3,1,1,1] => 1 [3,1,2] => 1 [3,2,1] => 1 [3,3] => 2 [4,1,1] => 1 [4,2] => 1 [5,1] => 1 [6] => 1 [1,1,1,1,1,1,1] => 7 [1,1,1,1,1,2] => 6 [1,1,1,1,2,1] => 5 [1,1,1,1,3] => 5 [1,1,1,2,1,1] => 4 [1,1,1,2,2] => 5 [1,1,1,3,1] => 4 [1,1,1,4] => 4 [1,1,2,1,1,1] => 3 [1,1,2,1,2] => 4 [1,1,2,2,1] => 4 [1,1,2,3] => 4 [1,1,3,1,1] => 3 [1,1,3,2] => 3 [1,1,4,1] => 3 [1,1,5] => 3 [1,2,1,1,1,1] => 2 [1,2,1,1,2] => 3 [1,2,1,2,1] => 3 [1,2,1,3] => 3 [1,2,2,1,1] => 3 [1,2,2,2] => 4 [1,2,3,1] => 3 [1,2,4] => 3 [1,3,1,1,1] => 2 [1,3,1,2] => 2 [1,3,2,1] => 2 [1,3,3] => 3 [1,4,1,1] => 2 [1,4,2] => 2 [1,5,1] => 2 [1,6] => 2 [2,1,1,1,1,1] => 1 [2,1,1,1,2] => 2 [2,1,1,2,1] => 2 [2,1,1,3] => 2 [2,1,2,1,1] => 2 [2,1,2,2] => 3 [2,1,3,1] => 2 [2,1,4] => 2 [2,2,1,1,1] => 2 [2,2,1,2] => 3 [2,2,2,1] => 3 [2,2,3] => 3 [2,3,1,1] => 2 [2,3,2] => 2 [2,4,1] => 2 [2,5] => 2 [3,1,1,1,1] => 1 [3,1,1,2] => 1 [3,1,2,1] => 1 [3,1,3] => 2 [3,2,1,1] => 1 [3,2,2] => 1 [3,3,1] => 2 [3,4] => 2 [4,1,1,1] => 1 [4,1,2] => 1 [4,2,1] => 1 [4,3] => 1 [5,1,1] => 1 [5,2] => 1 [6,1] => 1 [7] => 1 [1,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,2] => 7 [1,1,1,1,1,2,1] => 6 [1,1,1,1,1,3] => 6 [1,1,1,1,2,1,1] => 5 [1,1,1,1,2,2] => 6 [1,1,1,1,3,1] => 5 [1,1,1,1,4] => 5 [1,1,1,2,1,1,1] => 4 [1,1,1,2,1,2] => 5 [1,1,1,2,2,1] => 5 [1,1,1,2,3] => 5 [1,1,1,3,1,1] => 4 [1,1,1,3,2] => 4 [1,1,1,4,1] => 4 [1,1,1,5] => 4 [1,1,2,1,1,1,1] => 3 [1,1,2,1,1,2] => 4 [1,1,2,1,2,1] => 4 [1,1,2,1,3] => 4 [1,1,2,2,1,1] => 4 [1,1,2,2,2] => 5 [1,1,2,3,1] => 4 [1,1,2,4] => 4 [1,1,3,1,1,1] => 3 [1,1,3,1,2] => 3 [1,1,3,2,1] => 3 [1,1,3,3] => 4 [1,1,4,1,1] => 3 [1,1,4,2] => 3 [1,1,5,1] => 3 [1,1,6] => 3 [1,2,1,1,1,1,1] => 2 [1,2,1,1,1,2] => 3 [1,2,1,1,2,1] => 3 [1,2,1,1,3] => 3 [1,2,1,2,1,1] => 3 [1,2,1,2,2] => 4 [1,2,1,3,1] => 3 [1,2,1,4] => 3 [1,2,2,1,1,1] => 3 [1,2,2,1,2] => 4 [1,2,2,2,1] => 4 [1,2,2,3] => 4 [1,2,3,1,1] => 3 [1,2,3,2] => 3 [1,2,4,1] => 3 [1,2,5] => 3 [1,3,1,1,1,1] => 2 [1,3,1,1,2] => 2 [1,3,1,2,1] => 2 [1,3,1,3] => 3 [1,3,2,1,1] => 2 [1,3,2,2] => 2 [1,3,3,1] => 3 [1,3,4] => 3 [1,4,1,1,1] => 2 [1,4,1,2] => 2 [1,4,2,1] => 2 [1,4,3] => 2 [1,5,1,1] => 2 [1,5,2] => 2 [1,6,1] => 2 [1,7] => 2 [2,1,1,1,1,1,1] => 1 [2,1,1,1,1,2] => 2 [2,1,1,1,2,1] => 2 [2,1,1,1,3] => 2 [2,1,1,2,1,1] => 2 [2,1,1,2,2] => 3 [2,1,1,3,1] => 2 [2,1,1,4] => 2 [2,1,2,1,1,1] => 2 [2,1,2,1,2] => 3 [2,1,2,2,1] => 3 [2,1,2,3] => 3 [2,1,3,1,1] => 2 [2,1,3,2] => 2 [2,1,4,1] => 2 [2,1,5] => 2 [2,2,1,1,1,1] => 2 [2,2,1,1,2] => 3 [2,2,1,2,1] => 3 [2,2,1,3] => 3 [2,2,2,1,1] => 3 [2,2,2,2] => 4 [2,2,3,1] => 3 [2,2,4] => 3 [2,3,1,1,1] => 2 [2,3,1,2] => 2 [2,3,2,1] => 2 [2,3,3] => 3 [2,4,1,1] => 2 [2,4,2] => 2 [2,5,1] => 2 [2,6] => 2 [3,1,1,1,1,1] => 1 [3,1,1,1,2] => 1 [3,1,1,2,1] => 1 [3,1,1,3] => 2 [3,1,2,1,1] => 1 [3,1,2,2] => 1 [3,1,3,1] => 2 [3,1,4] => 2 [3,2,1,1,1] => 1 [3,2,1,2] => 1 [3,2,2,1] => 1 [3,2,3] => 2 [3,3,1,1] => 2 [3,3,2] => 2 [3,4,1] => 2 [3,5] => 2 [4,1,1,1,1] => 1 [4,1,1,2] => 1 [4,1,2,1] => 1 [4,1,3] => 1 [4,2,1,1] => 1 [4,2,2] => 1 [4,3,1] => 1 [4,4] => 2 [5,1,1,1] => 1 [5,1,2] => 1 [5,2,1] => 1 [5,3] => 1 [6,1,1] => 1 [6,2] => 1 [7,1] => 1 [8] => 1 [1,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,2] => 8 [1,1,1,1,1,1,2,1] => 7 [1,1,1,1,1,1,3] => 7 [1,1,1,1,1,2,1,1] => 6 [1,1,1,1,1,2,2] => 7 [1,1,1,1,1,3,1] => 6 [1,1,1,1,1,4] => 6 [1,1,1,1,2,1,1,1] => 5 [1,1,1,1,2,1,2] => 6 [1,1,1,1,2,2,1] => 6 [1,1,1,1,2,3] => 6 [1,1,1,1,3,1,1] => 5 [1,1,1,1,3,2] => 5 [1,1,1,1,4,1] => 5 [1,1,1,1,5] => 5 [1,1,1,2,1,1,1,1] => 4 [1,1,1,2,1,1,2] => 5 [1,1,1,2,1,2,1] => 5 [1,1,1,2,1,3] => 5 [1,1,1,2,2,1,1] => 5 [1,1,1,2,2,2] => 6 [1,1,1,2,3,1] => 5 [1,1,1,2,4] => 5 [1,1,1,3,1,1,1] => 4 [1,1,1,3,1,2] => 4 [1,1,1,3,2,1] => 4 [1,1,1,3,3] => 5 [1,1,1,4,1,1] => 4 [1,1,1,4,2] => 4 [1,1,1,5,1] => 4 [1,1,1,6] => 4 [1,1,2,1,1,1,1,1] => 3 [1,1,2,1,1,1,2] => 4 [1,1,2,1,1,2,1] => 4 [1,1,2,1,1,3] => 4 [1,1,2,1,2,1,1] => 4 [1,1,2,1,2,2] => 5 [1,1,2,1,3,1] => 4 [1,1,2,1,4] => 4 [1,1,2,2,1,1,1] => 4 [1,1,2,2,1,2] => 5 [1,1,2,2,2,1] => 5 [1,1,2,2,3] => 5 [1,1,2,3,1,1] => 4 [1,1,2,3,2] => 4 [1,1,2,4,1] => 4 [1,1,2,5] => 4 [1,1,3,1,1,1,1] => 3 [1,1,3,1,1,2] => 3 [1,1,3,1,2,1] => 3 [1,1,3,1,3] => 4 [1,1,3,2,1,1] => 3 [1,1,3,2,2] => 3 [1,1,3,3,1] => 4 [1,1,3,4] => 4 [1,1,4,1,1,1] => 3 [1,1,4,1,2] => 3 [1,1,4,2,1] => 3 [1,1,4,3] => 3 [1,1,5,1,1] => 3 [1,1,5,2] => 3 [1,1,6,1] => 3 [1,1,7] => 3 [1,2,1,1,1,1,1,1] => 2 [1,2,1,1,1,1,2] => 3 [1,2,1,1,1,2,1] => 3 [1,2,1,1,1,3] => 3 [1,2,1,1,2,1,1] => 3 [1,2,1,1,2,2] => 4 [1,2,1,1,3,1] => 3 [1,2,1,1,4] => 3 [1,2,1,2,1,1,1] => 3 [1,2,1,2,1,2] => 4 [1,2,1,2,2,1] => 4 [1,2,1,2,3] => 4 [1,2,1,3,1,1] => 3 [1,2,1,3,2] => 3 [1,2,1,4,1] => 3 [1,2,1,5] => 3 [1,2,2,1,1,1,1] => 3 [1,2,2,1,1,2] => 4 [1,2,2,1,2,1] => 4 [1,2,2,1,3] => 4 [1,2,2,2,1,1] => 4 [1,2,2,2,2] => 5 [1,2,2,3,1] => 4 [1,2,2,4] => 4 [1,2,3,1,1,1] => 3 [1,2,3,1,2] => 3 [1,2,3,2,1] => 3 [1,2,3,3] => 4 [1,2,4,1,1] => 3 [1,2,4,2] => 3 [1,2,5,1] => 3 [1,2,6] => 3 [1,3,1,1,1,1,1] => 2 [1,3,1,1,1,2] => 2 [1,3,1,1,2,1] => 2 [1,3,1,1,3] => 3 [1,3,1,2,1,1] => 2 [1,3,1,2,2] => 2 [1,3,1,3,1] => 3 [1,3,1,4] => 3 [1,3,2,1,1,1] => 2 [1,3,2,1,2] => 2 [1,3,2,2,1] => 2 [1,3,2,3] => 3 [1,3,3,1,1] => 3 [1,3,3,2] => 3 [1,3,4,1] => 3 [1,3,5] => 3 [1,4,1,1,1,1] => 2 [1,4,1,1,2] => 2 [1,4,1,2,1] => 2 [1,4,1,3] => 2 [1,4,2,1,1] => 2 [1,4,2,2] => 2 [1,4,3,1] => 2 [1,4,4] => 3 [1,5,1,1,1] => 2 [1,5,1,2] => 2 [1,5,2,1] => 2 [1,5,3] => 2 [1,6,1,1] => 2 [1,6,2] => 2 [1,7,1] => 2 [1,8] => 2 [2,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,2] => 2 [2,1,1,1,1,2,1] => 2 [2,1,1,1,1,3] => 2 [2,1,1,1,2,1,1] => 2 [2,1,1,1,2,2] => 3 [2,1,1,1,3,1] => 2 [2,1,1,1,4] => 2 [2,1,1,2,1,1,1] => 2 [2,1,1,2,1,2] => 3 [2,1,1,2,2,1] => 3 [2,1,1,2,3] => 3 [2,1,1,3,1,1] => 2 [2,1,1,3,2] => 2 [2,1,1,4,1] => 2 [2,1,1,5] => 2 [2,1,2,1,1,1,1] => 2 [2,1,2,1,1,2] => 3 [2,1,2,1,2,1] => 3 [2,1,2,1,3] => 3 [2,1,2,2,1,1] => 3 [2,1,2,2,2] => 4 [2,1,2,3,1] => 3 [2,1,2,4] => 3 [2,1,3,1,1,1] => 2 [2,1,3,1,2] => 2 [2,1,3,2,1] => 2 [2,1,3,3] => 3 [2,1,4,1,1] => 2 [2,1,4,2] => 2 [2,1,5,1] => 2 [2,1,6] => 2 [2,2,1,1,1,1,1] => 2 [2,2,1,1,1,2] => 3 [2,2,1,1,2,1] => 3 [2,2,1,1,3] => 3 [2,2,1,2,1,1] => 3 [2,2,1,2,2] => 4 [2,2,1,3,1] => 3 [2,2,1,4] => 3 [2,2,2,1,1,1] => 3 [2,2,2,1,2] => 4 [2,2,2,2,1] => 4 [2,2,2,3] => 4 [2,2,3,1,1] => 3 [2,2,3,2] => 3 [2,2,4,1] => 3 [2,2,5] => 3 [2,3,1,1,1,1] => 2 [2,3,1,1,2] => 2 [2,3,1,2,1] => 2 [2,3,1,3] => 3 [2,3,2,1,1] => 2 [2,3,2,2] => 2 [2,3,3,1] => 3 [2,3,4] => 3 [2,4,1,1,1] => 2 [2,4,1,2] => 2 [2,4,2,1] => 2 [2,4,3] => 2 [2,5,1,1] => 2 [2,5,2] => 2 [2,6,1] => 2 [2,7] => 2 [3,1,1,1,1,1,1] => 1 [3,1,1,1,1,2] => 1 [3,1,1,1,2,1] => 1 [3,1,1,1,3] => 2 [3,1,1,2,1,1] => 1 [3,1,1,2,2] => 1 [3,1,1,3,1] => 2 [3,1,1,4] => 2 [3,1,2,1,1,1] => 1 [3,1,2,1,2] => 1 [3,1,2,2,1] => 1 [3,1,2,3] => 2 [3,1,3,1,1] => 2 [3,1,3,2] => 2 [3,1,4,1] => 2 [3,1,5] => 2 [3,2,1,1,1,1] => 1 [3,2,1,1,2] => 1 [3,2,1,2,1] => 1 [3,2,1,3] => 2 [3,2,2,1,1] => 1 [3,2,2,2] => 1 [3,2,3,1] => 2 [3,2,4] => 2 [3,3,1,1,1] => 2 [3,3,1,2] => 2 [3,3,2,1] => 2 [3,3,3] => 3 [3,4,1,1] => 2 [3,4,2] => 2 [3,5,1] => 2 [3,6] => 2 [4,1,1,1,1,1] => 1 [4,1,1,1,2] => 1 [4,1,1,2,1] => 1 [4,1,1,3] => 1 [4,1,2,1,1] => 1 [4,1,2,2] => 1 [4,1,3,1] => 1 [4,1,4] => 2 [4,2,1,1,1] => 1 [4,2,1,2] => 1 [4,2,2,1] => 1 [4,2,3] => 1 [4,3,1,1] => 1 [4,3,2] => 1 [4,4,1] => 2 [4,5] => 2 [5,1,1,1,1] => 1 [5,1,1,2] => 1 [5,1,2,1] => 1 [5,1,3] => 1 [5,2,1,1] => 1 [5,2,2] => 1 [5,3,1] => 1 [5,4] => 1 [6,1,1,1] => 1 [6,1,2] => 1 [6,2,1] => 1 [6,3] => 1 [7,1,1] => 1 [7,2] => 1 [8,1] => 1 [9] => 1 ----------------------------------------------------------------------------- Created: Apr 08, 2017 at 21:58 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 19, 2017 at 22:06 by Christian Stump