***************************************************************************** * 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: St000548 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of different non-empty partial sums of an integer partition. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(l): return len(set(sum(l[i] for i in X) for X in Subsets(range(len(l)))))-1 ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 3 [1,1,1] => 3 [4] => 1 [3,1] => 3 [2,2] => 2 [2,1,1] => 4 [1,1,1,1] => 4 [5] => 1 [4,1] => 3 [3,2] => 3 [3,1,1] => 5 [2,2,1] => 5 [2,1,1,1] => 5 [1,1,1,1,1] => 5 [6] => 1 [5,1] => 3 [4,2] => 3 [4,1,1] => 5 [3,3] => 2 [3,2,1] => 6 [3,1,1,1] => 6 [2,2,2] => 3 [2,2,1,1] => 6 [2,1,1,1,1] => 6 [1,1,1,1,1,1] => 6 [7] => 1 [6,1] => 3 [5,2] => 3 [5,1,1] => 5 [4,3] => 3 [4,2,1] => 7 [4,1,1,1] => 7 [3,3,1] => 5 [3,2,2] => 5 [3,2,1,1] => 7 [3,1,1,1,1] => 7 [2,2,2,1] => 7 [2,2,1,1,1] => 7 [2,1,1,1,1,1] => 7 [1,1,1,1,1,1,1] => 7 [8] => 1 [7,1] => 3 [6,2] => 3 [6,1,1] => 5 [5,3] => 3 [5,2,1] => 7 [5,1,1,1] => 7 [4,4] => 2 [4,3,1] => 6 [4,2,2] => 4 [4,2,1,1] => 8 [4,1,1,1,1] => 8 [3,3,2] => 5 [3,3,1,1] => 8 [3,2,2,1] => 8 [3,2,1,1,1] => 8 [3,1,1,1,1,1] => 8 [2,2,2,2] => 4 [2,2,2,1,1] => 8 [2,2,1,1,1,1] => 8 [2,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1] => 8 [9] => 1 [8,1] => 3 [7,2] => 3 [7,1,1] => 5 [6,3] => 3 [6,2,1] => 7 [6,1,1,1] => 7 [5,4] => 3 [5,3,1] => 7 [5,2,2] => 5 [5,2,1,1] => 9 [5,1,1,1,1] => 9 [4,4,1] => 5 [4,3,2] => 7 [4,3,1,1] => 9 [4,2,2,1] => 9 [4,2,1,1,1] => 9 [4,1,1,1,1,1] => 9 [3,3,3] => 3 [3,3,2,1] => 9 [3,3,1,1,1] => 9 [3,2,2,2] => 7 [3,2,2,1,1] => 9 [3,2,1,1,1,1] => 9 [3,1,1,1,1,1,1] => 9 [2,2,2,2,1] => 9 [2,2,2,1,1,1] => 9 [2,2,1,1,1,1,1] => 9 [2,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1] => 9 [10] => 1 [9,1] => 3 [8,2] => 3 [8,1,1] => 5 [7,3] => 3 [7,2,1] => 7 [7,1,1,1] => 7 [6,4] => 3 [6,3,1] => 7 [6,2,2] => 5 [6,2,1,1] => 9 [6,1,1,1,1] => 9 [5,5] => 2 [5,4,1] => 6 [5,3,2] => 6 [5,3,1,1] => 10 [5,2,2,1] => 10 [5,2,1,1,1] => 10 [5,1,1,1,1,1] => 10 [4,4,2] => 5 [4,4,1,1] => 8 [4,3,3] => 5 [4,3,2,1] => 10 [4,3,1,1,1] => 10 [4,2,2,2] => 5 [4,2,2,1,1] => 10 [4,2,1,1,1,1] => 10 [4,1,1,1,1,1,1] => 10 [3,3,3,1] => 7 [3,3,2,2] => 8 [3,3,2,1,1] => 10 [3,3,1,1,1,1] => 10 [3,2,2,2,1] => 10 [3,2,2,1,1,1] => 10 [3,2,1,1,1,1,1] => 10 [3,1,1,1,1,1,1,1] => 10 [2,2,2,2,2] => 5 [2,2,2,2,1,1] => 10 [2,2,2,1,1,1,1] => 10 [2,2,1,1,1,1,1,1] => 10 [2,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1] => 10 [11] => 1 [10,1] => 3 [9,2] => 3 [9,1,1] => 5 [8,3] => 3 [8,2,1] => 7 [8,1,1,1] => 7 [7,4] => 3 [7,3,1] => 7 [7,2,2] => 5 [7,2,1,1] => 9 [7,1,1,1,1] => 9 [6,5] => 3 [6,4,1] => 7 [6,3,2] => 7 [6,3,1,1] => 11 [6,2,2,1] => 11 [6,2,1,1,1] => 11 [6,1,1,1,1,1] => 11 [5,5,1] => 5 [5,4,2] => 7 [5,4,1,1] => 9 [5,3,3] => 5 [5,3,2,1] => 11 [5,3,1,1,1] => 11 [5,2,2,2] => 7 [5,2,2,1,1] => 11 [5,2,1,1,1,1] => 11 [5,1,1,1,1,1,1] => 11 [4,4,3] => 5 [4,4,2,1] => 11 [4,4,1,1,1] => 11 [4,3,3,1] => 9 [4,3,2,2] => 9 [4,3,2,1,1] => 11 [4,3,1,1,1,1] => 11 [4,2,2,2,1] => 11 [4,2,2,1,1,1] => 11 [4,2,1,1,1,1,1] => 11 [4,1,1,1,1,1,1,1] => 11 [3,3,3,2] => 7 [3,3,3,1,1] => 11 [3,3,2,2,1] => 11 [3,3,2,1,1,1] => 11 [3,3,1,1,1,1,1] => 11 [3,2,2,2,2] => 9 [3,2,2,2,1,1] => 11 [3,2,2,1,1,1,1] => 11 [3,2,1,1,1,1,1,1] => 11 [3,1,1,1,1,1,1,1,1] => 11 [2,2,2,2,2,1] => 11 [2,2,2,2,1,1,1] => 11 [2,2,2,1,1,1,1,1] => 11 [2,2,1,1,1,1,1,1,1] => 11 [2,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1] => 11 [12] => 1 [11,1] => 3 [10,2] => 3 [10,1,1] => 5 [9,3] => 3 [9,2,1] => 7 [9,1,1,1] => 7 [8,4] => 3 [8,3,1] => 7 [8,2,2] => 5 [8,2,1,1] => 9 [8,1,1,1,1] => 9 [7,5] => 3 [7,4,1] => 7 [7,3,2] => 7 [7,3,1,1] => 11 [7,2,2,1] => 11 [7,2,1,1,1] => 11 [7,1,1,1,1,1] => 11 [6,6] => 2 [6,5,1] => 6 [6,4,2] => 6 [6,4,1,1] => 10 [6,3,3] => 4 [6,3,2,1] => 12 [6,3,1,1,1] => 12 [6,2,2,2] => 6 [6,2,2,1,1] => 12 [6,2,1,1,1,1] => 12 [6,1,1,1,1,1,1] => 12 [5,5,2] => 5 [5,5,1,1] => 8 [5,4,3] => 7 [5,4,2,1] => 12 [5,4,1,1,1] => 12 [5,3,3,1] => 10 [5,3,2,2] => 9 [5,3,2,1,1] => 12 [5,3,1,1,1,1] => 12 [5,2,2,2,1] => 12 [5,2,2,1,1,1] => 12 [5,2,1,1,1,1,1] => 12 [5,1,1,1,1,1,1,1] => 12 [4,4,4] => 3 [4,4,3,1] => 9 [4,4,2,2] => 6 [4,4,2,1,1] => 12 [4,4,1,1,1,1] => 12 [4,3,3,2] => 10 [4,3,3,1,1] => 12 [4,3,2,2,1] => 12 [4,3,2,1,1,1] => 12 [4,3,1,1,1,1,1] => 12 [4,2,2,2,2] => 6 [4,2,2,2,1,1] => 12 [4,2,2,1,1,1,1] => 12 [4,2,1,1,1,1,1,1] => 12 [4,1,1,1,1,1,1,1,1] => 12 [3,3,3,3] => 4 [3,3,3,2,1] => 12 [3,3,3,1,1,1] => 12 [3,3,2,2,2] => 10 [3,3,2,2,1,1] => 12 [3,3,2,1,1,1,1] => 12 [3,3,1,1,1,1,1,1] => 12 [3,2,2,2,2,1] => 12 [3,2,2,2,1,1,1] => 12 [3,2,2,1,1,1,1,1] => 12 [3,2,1,1,1,1,1,1,1] => 12 [3,1,1,1,1,1,1,1,1,1] => 12 [2,2,2,2,2,2] => 6 [2,2,2,2,2,1,1] => 12 [2,2,2,2,1,1,1,1] => 12 [2,2,2,1,1,1,1,1,1] => 12 [2,2,1,1,1,1,1,1,1,1] => 12 [2,1,1,1,1,1,1,1,1,1,1] => 12 [1,1,1,1,1,1,1,1,1,1,1,1] => 12 [8,5] => 3 [7,5,1] => 7 [7,4,2] => 7 [5,5,3] => 5 [5,4,4] => 5 [5,4,3,1] => 11 [5,4,2,2] => 9 [5,4,2,1,1] => 13 [5,4,1,1,1,1] => 13 [5,3,3,2] => 9 [5,3,3,1,1] => 13 [5,3,2,2,1] => 13 [5,3,2,1,1,1] => 13 [4,4,4,1] => 7 [4,4,3,2] => 11 [4,4,3,1,1] => 13 [4,4,2,2,1] => 13 [4,3,3,3] => 7 [4,3,3,2,1] => 13 [3,3,3,3,1] => 9 [3,3,3,2,2] => 11 [3,3,2,2,2,1] => 13 [9,5] => 3 [8,5,1] => 7 [7,5,2] => 6 [7,4,3] => 6 [6,4,4] => 5 [6,2,2,2,2] => 7 [5,5,4] => 5 [5,5,1,1,1,1] => 14 [5,4,3,2] => 12 [5,4,3,1,1] => 14 [5,4,2,2,1] => 14 [5,4,2,1,1,1] => 14 [5,3,3,2,1] => 14 [5,3,2,2,2] => 12 [5,2,2,2,2,1] => 14 [4,4,4,2] => 7 [4,4,3,3] => 8 [4,4,3,2,1] => 14 [4,3,2,2,2,1] => 14 [3,3,3,3,2] => 9 [3,3,3,3,1,1] => 14 [9,5,1] => 7 [8,5,2] => 7 [7,5,3] => 7 [6,5,4] => 7 [6,5,1,1,1,1] => 15 [6,3,3,3] => 5 [6,2,2,2,2,1] => 15 [5,5,5] => 3 [5,4,3,2,1] => 15 [5,4,3,1,1,1] => 15 [5,3,2,2,2,1] => 15 [4,4,4,3] => 7 [4,4,4,1,1,1] => 15 [3,3,3,3,3] => 5 [3,3,3,3,2,1] => 15 [8,5,3] => 6 [7,5,3,1] => 14 [5,5,3,3] => 8 [5,5,2,2,2] => 11 [5,4,3,2,1,1] => 16 [5,4,2,2,2,1] => 16 [4,4,4,4] => 4 [4,4,4,2,2] => 8 [4,3,3,3,2,1] => 16 [8,6,3] => 7 [6,5,3,3] => 9 [6,5,2,2,2] => 13 [6,4,4,3] => 11 [6,4,4,1,1,1] => 17 [6,3,3,3,2] => 11 [6,3,3,3,1,1] => 17 [5,5,4,3] => 11 [5,5,4,1,1,1] => 17 [5,5,2,2,2,1] => 17 [5,4,3,2,2,1] => 17 [5,3,3,3,2,1] => 17 [4,4,4,3,2] => 15 [4,4,4,3,1,1] => 17 [4,4,4,2,2,1] => 17 [4,4,4,3,2,1] => 18 [5,4,3,3,2,1] => 18 [6,3,3,3,2,1] => 18 [6,5,2,2,2,1] => 18 [5,5,3,3,1,1] => 18 [6,5,4,1,1,1] => 18 [5,5,3,3,2] => 13 [5,5,4,2,2] => 14 [6,4,4,2,2] => 9 [6,5,4,3] => 14 [9,6,3] => 6 [8,6,4] => 7 [5,4,4,3,2,1] => 19 [5,5,3,3,2,1] => 19 [5,5,4,2,2,1] => 19 [6,4,4,2,2,1] => 19 [5,5,4,3,1,1] => 19 [6,4,4,3,1,1] => 19 [6,5,3,3,1,1] => 19 [5,5,4,3,2] => 17 [6,4,4,3,2] => 17 [6,5,3,3,2] => 15 [6,5,4,2,2] => 15 [6,5,4,3,1] => 17 [6,5,4,1,1,1,1] => 19 [9,6,4] => 7 [8,5,4,2] => 15 [8,5,5,1] => 11 [5,5,4,3,2,1] => 20 [6,4,4,3,2,1] => 20 [6,5,3,3,2,1] => 20 [6,5,4,2,2,1] => 20 [6,5,4,3,1,1] => 20 [6,5,4,3,2] => 18 [6,5,2,2,2,2,1] => 20 [6,5,4,2,1,1,1] => 20 [7,5,4,3,1] => 18 [8,6,4,2] => 10 [10,6,4] => 6 [10,7,3] => 6 [9,7,4] => 7 [9,5,5,1] => 10 [6,5,4,3,2,1] => 21 [6,3,3,3,3,2,1] => 21 [6,5,3,2,2,2,1] => 21 [6,5,4,3,1,1,1] => 21 [11,7,3] => 7 [4,4,4,4,3,2,1] => 22 [6,4,3,3,3,2,1] => 22 [6,5,4,2,2,2,1] => 22 [6,5,4,3,2,1,1] => 22 [9,6,4,3] => 13 [5,4,4,4,3,2,1] => 23 [6,5,3,3,3,2,1] => 23 [6,5,4,3,2,2,1] => 23 [9,6,5,3] => 13 [8,6,5,3,1] => 21 [6,4,4,4,3,2,1] => 24 [6,5,4,3,3,2,1] => 24 [11,7,5,1] => 14 [9,7,5,3] => 14 [5,5,5,4,3,2,1] => 25 [6,5,4,4,3,2,1] => 25 [9,7,5,3,1] => 23 [10,7,5,3] => 13 [6,5,5,4,3,2,1] => 26 [9,7,5,4,1] => 22 [6,6,5,4,3,2,1] => 27 [7,6,5,4,3,2] => 25 [7,6,5,4,3,2,1] => 28 [7,6,5,4,3,1,1,1] => 28 [10,7,6,4,1] => 22 [9,7,6,4,2] => 21 [10,8,5,4,1] => 22 [7,6,5,4,3,2,1,1] => 29 [7,6,5,4,2,2,2,1] => 29 [10,8,6,4,1] => 25 [9,7,5,5,3,1] => 28 [7,6,5,4,3,2,2,1] => 30 [7,6,5,3,3,3,2,1] => 30 [11,8,6,4,1] => 26 [10,8,6,4,2] => 15 [7,6,5,4,3,3,2,1] => 31 [7,6,4,4,4,3,2,1] => 31 [11,8,6,5,1] => 23 [7,6,5,4,4,3,2,1] => 32 [7,5,5,5,4,3,2,1] => 32 [7,6,5,5,4,3,2,1] => 33 [6,6,6,5,4,3,2,1] => 33 [7,6,6,5,4,3,2,1] => 34 [12,9,7,5,1] => 26 [7,7,6,5,4,3,2,1] => 35 [13,9,7,5,1] => 27 [11,9,7,5,3,1] => 34 [11,8,7,5,4,1] => 32 [8,7,6,5,4,3,2,1] => 36 [8,7,6,5,4,3,2,1,1] => 37 [8,7,6,5,4,3,2,2,1] => 38 [8,7,6,5,4,3,3,2,1] => 39 [8,7,6,5,4,4,3,2,1] => 40 [11,9,7,5,5,3] => 32 [8,7,6,5,5,4,3,2,1] => 41 [8,7,6,6,5,4,3,2,1] => 42 [8,7,7,6,5,4,3,2,1] => 43 [8,8,7,6,5,4,3,2,1] => 44 [9,8,7,6,5,4,3,2,1] => 45 [11,9,7,7,5,3,3] => 39 [11,9,7,6,5,3,1] => 40 [13,11,9,7,5,3,1] => 47 [13,11,9,7,7,5,3,1] => 54 [17,13,11,9,7,5,1] => 57 [15,13,11,9,7,5,3,1] => 62 [29,23,19,17,13,11,7,1] => 98 ----------------------------------------------------------------------------- Created: Jul 16, 2016 at 16:39 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 17, 2021 at 10:53 by Martin Rubey