***************************************************************************** * 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: St000346 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of coarsenings of a partition. A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$. ----------------------------------------------------------------------------- References: [1] Birkhoff, G. Lattice theory [[MathSciNet:0598630]] [2] Ziegler, Günter M. On the poset of partitions of an integer [[MathSciNet:0847552]] [3] Perry, J. M. Counting refinements of partitions [[MathOverflow:226656]] [4] [[St000345]] ----------------------------------------------------------------------------- Code: @cached_function def PartitionPoset(n): return posets.IntegerPartitions(n) def statistic(part): P = PartitionPoset(sum(part)) return len(P.order_ideal([tuple(part)])) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 2 [1,1,1] => 3 [4] => 1 [3,1] => 2 [2,2] => 2 [2,1,1] => 4 [1,1,1,1] => 5 [5] => 1 [4,1] => 2 [3,2] => 2 [3,1,1] => 4 [2,2,1] => 4 [2,1,1,1] => 6 [1,1,1,1,1] => 7 [6] => 1 [5,1] => 2 [4,2] => 2 [4,1,1] => 4 [3,3] => 2 [3,2,1] => 5 [3,1,1,1] => 7 [2,2,2] => 3 [2,2,1,1] => 8 [2,1,1,1,1] => 10 [1,1,1,1,1,1] => 11 [7] => 1 [6,1] => 2 [5,2] => 2 [5,1,1] => 4 [4,3] => 2 [4,2,1] => 5 [4,1,1,1] => 7 [3,3,1] => 4 [3,2,2] => 4 [3,2,1,1] => 9 [3,1,1,1,1] => 11 [2,2,2,1] => 7 [2,2,1,1,1] => 12 [2,1,1,1,1,1] => 14 [1,1,1,1,1,1,1] => 15 [8] => 1 [7,1] => 2 [6,2] => 2 [6,1,1] => 4 [5,3] => 2 [5,2,1] => 5 [5,1,1,1] => 7 [4,4] => 2 [4,3,1] => 5 [4,2,2] => 4 [4,2,1,1] => 10 [4,1,1,1,1] => 12 [3,3,2] => 4 [3,3,1,1] => 9 [3,2,2,1] => 10 [3,2,1,1,1] => 15 [3,1,1,1,1,1] => 17 [2,2,2,2] => 5 [2,2,2,1,1] => 14 [2,2,1,1,1,1] => 19 [2,1,1,1,1,1,1] => 21 [1,1,1,1,1,1,1,1] => 22 [9] => 1 [8,1] => 2 [7,2] => 2 [7,1,1] => 4 [6,3] => 2 [6,2,1] => 5 [6,1,1,1] => 7 [5,4] => 2 [5,3,1] => 5 [5,2,2] => 4 [5,2,1,1] => 10 [5,1,1,1,1] => 12 [4,4,1] => 4 [4,3,2] => 5 [4,3,1,1] => 10 [4,2,2,1] => 10 [4,2,1,1,1] => 16 [4,1,1,1,1,1] => 18 [3,3,3] => 3 [3,3,2,1] => 10 [3,3,1,1,1] => 15 [3,2,2,2] => 7 [3,2,2,1,1] => 18 [3,2,1,1,1,1] => 23 [3,1,1,1,1,1,1] => 25 [2,2,2,2,1] => 12 [2,2,2,1,1,1] => 22 [2,2,1,1,1,1,1] => 27 [2,1,1,1,1,1,1,1] => 29 [1,1,1,1,1,1,1,1,1] => 30 [10] => 1 [9,1] => 2 [8,2] => 2 [8,1,1] => 4 [7,3] => 2 [7,2,1] => 5 [7,1,1,1] => 7 [6,4] => 2 [6,3,1] => 5 [6,2,2] => 4 [6,2,1,1] => 10 [6,1,1,1,1] => 12 [5,5] => 2 [5,4,1] => 5 [5,3,2] => 5 [5,3,1,1] => 11 [5,2,2,1] => 11 [5,2,1,1,1] => 17 [5,1,1,1,1,1] => 19 [4,4,2] => 4 [4,4,1,1] => 9 [4,3,3] => 4 [4,3,2,1] => 13 [4,3,1,1,1] => 18 [4,2,2,2] => 6 [4,2,2,1,1] => 20 [4,2,1,1,1,1] => 26 [4,1,1,1,1,1,1] => 28 [3,3,3,1] => 7 [3,3,2,2] => 9 [3,3,2,1,1] => 20 [3,3,1,1,1,1] => 25 [3,2,2,2,1] => 18 [3,2,2,1,1,1] => 29 [3,2,1,1,1,1,1] => 34 [3,1,1,1,1,1,1,1] => 36 [2,2,2,2,2] => 7 [2,2,2,2,1,1] => 24 [2,2,2,1,1,1,1] => 34 [2,2,1,1,1,1,1,1] => 39 [2,1,1,1,1,1,1,1,1] => 41 [1,1,1,1,1,1,1,1,1,1] => 42 [11] => 1 [10,1] => 2 [9,2] => 2 [9,1,1] => 4 [8,3] => 2 [8,2,1] => 5 [8,1,1,1] => 7 [7,4] => 2 [7,3,1] => 5 [7,2,2] => 4 [7,2,1,1] => 10 [7,1,1,1,1] => 12 [6,5] => 2 [6,4,1] => 5 [6,3,2] => 5 [6,3,1,1] => 11 [6,2,2,1] => 11 [6,2,1,1,1] => 17 [6,1,1,1,1,1] => 19 [5,5,1] => 4 [5,4,2] => 5 [5,4,1,1] => 10 [5,3,3] => 4 [5,3,2,1] => 13 [5,3,1,1,1] => 19 [5,2,2,2] => 7 [5,2,2,1,1] => 21 [5,2,1,1,1,1] => 27 [5,1,1,1,1,1,1] => 29 [4,4,3] => 4 [4,4,2,1] => 11 [4,4,1,1,1] => 16 [4,3,3,1] => 10 [4,3,2,2] => 10 [4,3,2,1,1] => 24 [4,3,1,1,1,1] => 29 [4,2,2,2,1] => 17 [4,2,2,1,1,1] => 32 [4,2,1,1,1,1,1] => 38 [4,1,1,1,1,1,1,1] => 40 [3,3,3,2] => 7 [3,3,3,1,1] => 16 [3,3,2,2,1] => 21 [3,3,2,1,1,1] => 32 [3,3,1,1,1,1,1] => 37 [3,2,2,2,2] => 12 [3,2,2,2,1,1] => 32 [3,2,2,1,1,1,1] => 43 [3,2,1,1,1,1,1,1] => 48 [3,1,1,1,1,1,1,1,1] => 50 [2,2,2,2,2,1] => 19 [2,2,2,2,1,1,1] => 38 [2,2,2,1,1,1,1,1] => 48 [2,2,1,1,1,1,1,1,1] => 53 [2,1,1,1,1,1,1,1,1,1] => 55 [1,1,1,1,1,1,1,1,1,1,1] => 56 [12] => 1 [11,1] => 2 [10,2] => 2 [10,1,1] => 4 [9,3] => 2 [9,2,1] => 5 [9,1,1,1] => 7 [8,4] => 2 [8,3,1] => 5 [8,2,2] => 4 [8,2,1,1] => 10 [8,1,1,1,1] => 12 [7,5] => 2 [7,4,1] => 5 [7,3,2] => 5 [7,3,1,1] => 11 [7,2,2,1] => 11 [7,2,1,1,1] => 17 [7,1,1,1,1,1] => 19 [6,6] => 2 [6,5,1] => 5 [6,4,2] => 5 [6,4,1,1] => 11 [6,3,3] => 4 [6,3,2,1] => 14 [6,3,1,1,1] => 20 [6,2,2,2] => 7 [6,2,2,1,1] => 22 [6,2,1,1,1,1] => 28 [6,1,1,1,1,1,1] => 30 [5,5,2] => 4 [5,5,1,1] => 9 [5,4,3] => 5 [5,4,2,1] => 14 [5,4,1,1,1] => 19 [5,3,3,1] => 11 [5,3,2,2] => 10 [5,3,2,1,1] => 26 [5,3,1,1,1,1] => 32 [5,2,2,2,1] => 20 [5,2,2,1,1,1] => 35 [5,2,1,1,1,1,1] => 41 [5,1,1,1,1,1,1,1] => 43 [4,4,4] => 3 [4,4,3,1] => 10 [4,4,2,2] => 8 [4,4,2,1,1] => 23 [4,4,1,1,1,1] => 28 [4,3,3,2] => 11 [4,3,3,1,1] => 22 [4,3,2,2,1] => 26 [4,3,2,1,1,1] => 40 [4,3,1,1,1,1,1] => 45 [4,2,2,2,2] => 10 [4,2,2,2,1,1] => 35 [4,2,2,1,1,1,1] => 50 [4,2,1,1,1,1,1,1] => 56 [4,1,1,1,1,1,1,1,1] => 58 [3,3,3,3] => 5 [3,3,3,2,1] => 19 [3,3,3,1,1,1] => 29 [3,3,2,2,2] => 16 [3,3,2,2,1,1] => 40 [3,3,2,1,1,1,1] => 51 [3,3,1,1,1,1,1,1] => 56 [3,2,2,2,2,1] => 31 [3,2,2,2,1,1,1] => 52 [3,2,2,1,1,1,1,1] => 63 [3,2,1,1,1,1,1,1,1] => 68 [3,1,1,1,1,1,1,1,1,1] => 70 [2,2,2,2,2,2] => 11 [2,2,2,2,2,1,1] => 39 [2,2,2,2,1,1,1,1] => 59 [2,2,2,1,1,1,1,1,1] => 69 [2,2,1,1,1,1,1,1,1,1] => 74 [2,1,1,1,1,1,1,1,1,1,1] => 76 [1,1,1,1,1,1,1,1,1,1,1,1] => 77 ----------------------------------------------------------------------------- Created: Dec 23, 2015 at 16:30 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 20:59 by Martin Rubey