***************************************************************************** * 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: St000913 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of ways to refine the partition into singletons. For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$. In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(la): P = posets.IntegerPartitions(la.size()) H = P.hasse_diagram() e = H.vertices()[0] f = tuple(la) return len(H.all_simple_paths([f], [e], trivial=True)) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 1 [1,1,1] => 1 [4] => 2 [3,1] => 1 [2,2] => 1 [2,1,1] => 1 [1,1,1,1] => 1 [5] => 4 [4,1] => 2 [3,2] => 2 [3,1,1] => 1 [2,2,1] => 1 [2,1,1,1] => 1 [1,1,1,1,1] => 1 [6] => 11 [5,1] => 4 [4,2] => 5 [4,1,1] => 2 [3,3] => 2 [3,2,1] => 2 [3,1,1,1] => 1 [2,2,2] => 1 [2,2,1,1] => 1 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 1 [7] => 33 [6,1] => 11 [5,2] => 12 [5,1,1] => 4 [4,3] => 10 [4,2,1] => 5 [4,1,1,1] => 2 [3,3,1] => 2 [3,2,2] => 3 [3,2,1,1] => 2 [3,1,1,1,1] => 1 [2,2,2,1] => 1 [2,2,1,1,1] => 1 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 1 [8] => 116 [7,1] => 33 [6,2] => 37 [6,1,1] => 11 [5,3] => 27 [5,2,1] => 12 [5,1,1,1] => 4 [4,4] => 19 [4,3,1] => 10 [4,2,2] => 9 [4,2,1,1] => 5 [4,1,1,1,1] => 2 [3,3,2] => 5 [3,3,1,1] => 2 [3,2,2,1] => 3 [3,2,1,1,1] => 2 [3,1,1,1,1,1] => 1 [2,2,2,2] => 1 [2,2,2,1,1] => 1 [2,2,1,1,1,1] => 1 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 1 [9] => 435 [8,1] => 116 [7,2] => 123 [7,1,1] => 33 [6,3] => 97 [6,2,1] => 37 [6,1,1,1] => 11 [5,4] => 99 [5,3,1] => 27 [5,2,2] => 25 [5,2,1,1] => 12 [5,1,1,1,1] => 4 [4,4,1] => 19 [4,3,2] => 28 [4,3,1,1] => 10 [4,2,2,1] => 9 [4,2,1,1,1] => 5 [4,1,1,1,1,1] => 2 [3,3,3] => 5 [3,3,2,1] => 5 [3,3,1,1,1] => 2 [3,2,2,2] => 4 [3,2,2,1,1] => 3 [3,2,1,1,1,1] => 2 [3,1,1,1,1,1,1] => 1 [2,2,2,2,1] => 1 [2,2,2,1,1,1] => 1 [2,2,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1832 [9,1] => 435 [8,2] => 474 [8,1,1] => 116 [7,3] => 351 [7,2,1] => 123 [7,1,1,1] => 33 [6,4] => 384 [6,3,1] => 97 [6,2,2] => 85 [6,2,1,1] => 37 [6,1,1,1,1] => 11 [5,5] => 188 [5,4,1] => 99 [5,3,2] => 89 [5,3,1,1] => 27 [5,2,2,1] => 25 [5,2,1,1,1] => 12 [5,1,1,1,1,1] => 4 [4,4,2] => 61 [4,4,1,1] => 19 [4,3,3] => 42 [4,3,2,1] => 28 [4,3,1,1,1] => 10 [4,2,2,2] => 14 [4,2,2,1,1] => 9 [4,2,1,1,1,1] => 5 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 5 [3,3,2,2] => 9 [3,3,2,1,1] => 5 [3,3,1,1,1,1] => 2 [3,2,2,2,1] => 4 [3,2,2,1,1,1] => 3 [3,2,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1] => 1 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 1 [2,2,2,1,1,1,1] => 1 [2,2,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 8167 [10,1] => 1832 [9,2] => 1907 [9,1,1] => 435 [8,3] => 1470 [8,2,1] => 474 [8,1,1,1] => 116 [7,4] => 1551 [7,3,1] => 351 [7,2,2] => 308 [7,2,1,1] => 123 [7,1,1,1,1] => 33 [6,5] => 1407 [6,4,1] => 384 [6,3,2] => 341 [6,3,1,1] => 97 [6,2,2,1] => 85 [6,2,1,1,1] => 37 [6,1,1,1,1,1] => 11 [5,5,1] => 188 [5,4,2] => 349 [5,4,1,1] => 99 [5,3,3] => 145 [5,3,2,1] => 89 [5,3,1,1,1] => 27 [5,2,2,2] => 44 [5,2,2,1,1] => 25 [5,2,1,1,1,1] => 12 [5,1,1,1,1,1,1] => 4 [4,4,3] => 159 [4,4,2,1] => 61 [4,4,1,1,1] => 19 [4,3,3,1] => 42 [4,3,2,2] => 56 [4,3,2,1,1] => 28 [4,3,1,1,1,1] => 10 [4,2,2,2,1] => 14 [4,2,2,1,1,1] => 9 [4,2,1,1,1,1,1] => 5 [4,1,1,1,1,1,1,1] => 2 [3,3,3,2] => 14 [3,3,3,1,1] => 5 [3,3,2,2,1] => 9 [3,3,2,1,1,1] => 5 [3,3,1,1,1,1,1] => 2 [3,2,2,2,2] => 5 [3,2,2,2,1,1] => 4 [3,2,2,1,1,1,1] => 3 [3,2,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,1] => 1 [2,2,2,2,1,1,1] => 1 [2,2,2,1,1,1,1,1] => 1 [2,2,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 39700 [11,1] => 8167 [10,2] => 8593 [10,1,1] => 1832 [9,3] => 6314 [9,2,1] => 1907 [9,1,1,1] => 435 [8,4] => 7084 [8,3,1] => 1470 [8,2,2] => 1285 [8,2,1,1] => 474 [8,1,1,1,1] => 116 [7,5] => 6009 [7,4,1] => 1551 [7,3,2] => 1329 [7,3,1,1] => 351 [7,2,2,1] => 308 [7,2,1,1,1] => 123 [7,1,1,1,1,1] => 33 [6,6] => 3533 [6,5,1] => 1407 [6,4,2] => 1500 [6,4,1,1] => 384 [6,3,3] => 626 [6,3,2,1] => 341 [6,3,1,1,1] => 97 [6,2,2,2] => 163 [6,2,2,1,1] => 85 [6,2,1,1,1,1] => 37 [6,1,1,1,1,1,1] => 11 [5,5,2] => 740 [5,5,1,1] => 188 [5,4,3] => 982 [5,4,2,1] => 349 [5,4,1,1,1] => 99 [5,3,3,1] => 145 [5,3,2,2] => 203 [5,3,2,1,1] => 89 [5,3,1,1,1,1] => 27 [5,2,2,2,1] => 44 [5,2,2,1,1,1] => 25 [5,2,1,1,1,1,1] => 12 [5,1,1,1,1,1,1,1] => 4 [4,4,4] => 296 [4,4,3,1] => 159 [4,4,2,2] => 137 [4,4,2,1,1] => 61 [4,4,1,1,1,1] => 19 [4,3,3,2] => 126 [4,3,3,1,1] => 42 [4,3,2,2,1] => 56 [4,3,2,1,1,1] => 28 [4,3,1,1,1,1,1] => 10 [4,2,2,2,2] => 20 [4,2,2,2,1,1] => 14 [4,2,2,1,1,1,1] => 9 [4,2,1,1,1,1,1,1] => 5 [4,1,1,1,1,1,1,1,1] => 2 [3,3,3,3] => 14 [3,3,3,2,1] => 14 [3,3,3,1,1,1] => 5 [3,3,2,2,2] => 14 [3,3,2,2,1,1] => 9 [3,3,2,1,1,1,1] => 5 [3,3,1,1,1,1,1,1] => 2 [3,2,2,2,2,1] => 5 [3,2,2,2,1,1,1] => 4 [3,2,2,1,1,1,1,1] => 3 [3,2,1,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 1 [2,2,2,2,1,1,1,1] => 1 [2,2,2,1,1,1,1,1,1] => 1 [2,2,1,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [5,4,3,1] => 982 [5,4,2,2] => 855 [5,4,2,1,1] => 349 [5,3,3,2] => 502 [5,3,3,1,1] => 145 [5,3,2,2,1] => 203 [4,4,3,2] => 518 [4,4,3,1,1] => 159 [4,4,2,2,1] => 137 [4,3,3,2,1] => 126 [5,4,3,2] => 3516 [5,4,3,1,1] => 982 [5,4,2,2,1] => 855 [5,3,3,2,1] => 502 [4,4,3,2,1] => 518 [5,4,3,2,1] => 3516 ----------------------------------------------------------------------------- Created: Jul 19, 2017 at 15:58 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jan 17, 2018 at 23:29 by Martin Rubey