***************************************************************************** * 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: St000278 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. ----------------------------------------------------------------------------- References: [1] Preferred multisets: triangle of numbers refining A007318 using format described in A036038. [[OEIS:A048996]] ----------------------------------------------------------------------------- Code: def statistic(la): return multinomial(la.to_exp()) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 2 [1,1,1] => 1 [4] => 1 [3,1] => 2 [2,2] => 1 [2,1,1] => 3 [1,1,1,1] => 1 [5] => 1 [4,1] => 2 [3,2] => 2 [3,1,1] => 3 [2,2,1] => 3 [2,1,1,1] => 4 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 2 [4,2] => 2 [4,1,1] => 3 [3,3] => 1 [3,2,1] => 6 [3,1,1,1] => 4 [2,2,2] => 1 [2,2,1,1] => 6 [2,1,1,1,1] => 5 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 2 [5,2] => 2 [5,1,1] => 3 [4,3] => 2 [4,2,1] => 6 [4,1,1,1] => 4 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 12 [3,1,1,1,1] => 5 [2,2,2,1] => 4 [2,2,1,1,1] => 10 [2,1,1,1,1,1] => 6 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 2 [6,2] => 2 [6,1,1] => 3 [5,3] => 2 [5,2,1] => 6 [5,1,1,1] => 4 [4,4] => 1 [4,3,1] => 6 [4,2,2] => 3 [4,2,1,1] => 12 [4,1,1,1,1] => 5 [3,3,2] => 3 [3,3,1,1] => 6 [3,2,2,1] => 12 [3,2,1,1,1] => 20 [3,1,1,1,1,1] => 6 [2,2,2,2] => 1 [2,2,2,1,1] => 10 [2,2,1,1,1,1] => 15 [2,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 2 [7,2] => 2 [7,1,1] => 3 [6,3] => 2 [6,2,1] => 6 [6,1,1,1] => 4 [5,4] => 2 [5,3,1] => 6 [5,2,2] => 3 [5,2,1,1] => 12 [5,1,1,1,1] => 5 [4,4,1] => 3 [4,3,2] => 6 [4,3,1,1] => 12 [4,2,2,1] => 12 [4,2,1,1,1] => 20 [4,1,1,1,1,1] => 6 [3,3,3] => 1 [3,3,2,1] => 12 [3,3,1,1,1] => 10 [3,2,2,2] => 4 [3,2,2,1,1] => 30 [3,2,1,1,1,1] => 30 [3,1,1,1,1,1,1] => 7 [2,2,2,2,1] => 5 [2,2,2,1,1,1] => 20 [2,2,1,1,1,1,1] => 21 [2,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 2 [8,2] => 2 [8,1,1] => 3 [7,3] => 2 [7,2,1] => 6 [7,1,1,1] => 4 [6,4] => 2 [6,3,1] => 6 [6,2,2] => 3 [6,2,1,1] => 12 [6,1,1,1,1] => 5 [5,5] => 1 [5,4,1] => 6 [5,3,2] => 6 [5,3,1,1] => 12 [5,2,2,1] => 12 [5,2,1,1,1] => 20 [5,1,1,1,1,1] => 6 [4,4,2] => 3 [4,4,1,1] => 6 [4,3,3] => 3 [4,3,2,1] => 24 [4,3,1,1,1] => 20 [4,2,2,2] => 4 [4,2,2,1,1] => 30 [4,2,1,1,1,1] => 30 [4,1,1,1,1,1,1] => 7 [3,3,3,1] => 4 [3,3,2,2] => 6 [3,3,2,1,1] => 30 [3,3,1,1,1,1] => 15 [3,2,2,2,1] => 20 [3,2,2,1,1,1] => 60 [3,2,1,1,1,1,1] => 42 [3,1,1,1,1,1,1,1] => 8 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 15 [2,2,2,1,1,1,1] => 35 [2,2,1,1,1,1,1,1] => 28 [2,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 2 [9,2] => 2 [9,1,1] => 3 [8,3] => 2 [8,2,1] => 6 [8,1,1,1] => 4 [7,4] => 2 [7,3,1] => 6 [7,2,2] => 3 [7,2,1,1] => 12 [7,1,1,1,1] => 5 [6,5] => 2 [6,4,1] => 6 [6,3,2] => 6 [6,3,1,1] => 12 [6,2,2,1] => 12 [6,2,1,1,1] => 20 [6,1,1,1,1,1] => 6 [5,5,1] => 3 [5,4,2] => 6 [5,4,1,1] => 12 [5,3,3] => 3 [5,3,2,1] => 24 [5,3,1,1,1] => 20 [5,2,2,2] => 4 [5,2,2,1,1] => 30 [5,2,1,1,1,1] => 30 [5,1,1,1,1,1,1] => 7 [4,4,3] => 3 [4,4,2,1] => 12 [4,4,1,1,1] => 10 [4,3,3,1] => 12 [4,3,2,2] => 12 [4,3,2,1,1] => 60 [4,3,1,1,1,1] => 30 [4,2,2,2,1] => 20 [4,2,2,1,1,1] => 60 [4,2,1,1,1,1,1] => 42 [4,1,1,1,1,1,1,1] => 8 [3,3,3,2] => 4 [3,3,3,1,1] => 10 [3,3,2,2,1] => 30 [3,3,2,1,1,1] => 60 [3,3,1,1,1,1,1] => 21 [3,2,2,2,2] => 5 [3,2,2,2,1,1] => 60 [3,2,2,1,1,1,1] => 105 [3,2,1,1,1,1,1,1] => 56 [3,1,1,1,1,1,1,1,1] => 9 [2,2,2,2,2,1] => 6 [2,2,2,2,1,1,1] => 35 [2,2,2,1,1,1,1,1] => 56 [2,2,1,1,1,1,1,1,1] => 36 [2,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 2 [10,2] => 2 [10,1,1] => 3 [9,3] => 2 [9,2,1] => 6 [9,1,1,1] => 4 [8,4] => 2 [8,3,1] => 6 [8,2,2] => 3 [8,2,1,1] => 12 [8,1,1,1,1] => 5 [7,5] => 2 [7,4,1] => 6 [7,3,2] => 6 [7,3,1,1] => 12 [7,2,2,1] => 12 [7,2,1,1,1] => 20 [7,1,1,1,1,1] => 6 [6,6] => 1 [6,5,1] => 6 [6,4,2] => 6 [6,4,1,1] => 12 [6,3,3] => 3 [6,3,2,1] => 24 [6,3,1,1,1] => 20 [6,2,2,2] => 4 [6,2,2,1,1] => 30 [6,2,1,1,1,1] => 30 [6,1,1,1,1,1,1] => 7 [5,5,2] => 3 [5,5,1,1] => 6 [5,4,3] => 6 [5,4,2,1] => 24 [5,4,1,1,1] => 20 [5,3,3,1] => 12 [5,3,2,2] => 12 [5,3,2,1,1] => 60 [5,3,1,1,1,1] => 30 [5,2,2,2,1] => 20 [5,2,2,1,1,1] => 60 [5,2,1,1,1,1,1] => 42 [5,1,1,1,1,1,1,1] => 8 [4,4,4] => 1 [4,4,3,1] => 12 [4,4,2,2] => 6 [4,4,2,1,1] => 30 [4,4,1,1,1,1] => 15 [4,3,3,2] => 12 [4,3,3,1,1] => 30 [4,3,2,2,1] => 60 [4,3,2,1,1,1] => 120 [4,3,1,1,1,1,1] => 42 [4,2,2,2,2] => 5 [4,2,2,2,1,1] => 60 [4,2,2,1,1,1,1] => 105 [4,2,1,1,1,1,1,1] => 56 [4,1,1,1,1,1,1,1,1] => 9 [3,3,3,3] => 1 [3,3,3,2,1] => 20 [3,3,3,1,1,1] => 20 [3,3,2,2,2] => 10 [3,3,2,2,1,1] => 90 [3,3,2,1,1,1,1] => 105 [3,3,1,1,1,1,1,1] => 28 [3,2,2,2,2,1] => 30 [3,2,2,2,1,1,1] => 140 [3,2,2,1,1,1,1,1] => 168 [3,2,1,1,1,1,1,1,1] => 72 [3,1,1,1,1,1,1,1,1,1] => 10 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 21 [2,2,2,2,1,1,1,1] => 70 [2,2,2,1,1,1,1,1,1] => 84 [2,2,1,1,1,1,1,1,1,1] => 45 [2,1,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Sep 11, 2015 at 22:04 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 20:49 by Martin Rubey