***************************************************************************** * 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: St000515 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of invariant set partitions when acting with a permutation of given cycle type. ----------------------------------------------------------------------------- References: [1] Bergeron, F., Labelle, G., Leroux, P. Combinatorial species and tree-like structures [[MathSciNet:1629341]] ----------------------------------------------------------------------------- Code: def statistic(la): Partitionspecies = species.PartitionSpecies().cycle_index_series() return Partitionspecies.count(la) ----------------------------------------------------------------------------- Statistic values: [2] => 2 [1,1] => 2 [3] => 2 [2,1] => 3 [1,1,1] => 5 [4] => 3 [3,1] => 3 [2,2] => 7 [2,1,1] => 7 [1,1,1,1] => 15 [5] => 2 [4,1] => 4 [3,2] => 5 [3,1,1] => 7 [2,2,1] => 12 [2,1,1,1] => 20 [1,1,1,1,1] => 52 [6] => 4 [5,1] => 3 [4,2] => 9 [4,1,1] => 9 [3,3] => 8 [3,2,1] => 10 [3,1,1,1] => 20 [2,2,2] => 31 [2,2,1,1] => 31 [2,1,1,1,1] => 67 [1,1,1,1,1,1] => 203 [7] => 2 [6,1] => 5 [5,2] => 5 [5,1,1] => 7 [4,3] => 7 [4,2,1] => 15 [4,1,1,1] => 25 [3,3,1] => 13 [3,2,2] => 19 [3,2,1,1] => 27 [3,1,1,1,1] => 67 [2,2,2,1] => 59 [2,2,1,1,1] => 97 [2,1,1,1,1,1] => 255 [1,1,1,1,1,1,1] => 877 [8] => 4 [7,1] => 3 [6,2] => 11 [6,1,1] => 11 [5,3] => 5 [5,2,1] => 10 [5,1,1,1] => 20 [4,4] => 16 [4,3,1] => 13 [4,2,2] => 38 [4,2,1,1] => 38 [4,1,1,1,1] => 82 [3,3,2] => 21 [3,3,1,1] => 33 [3,2,2,1] => 43 [3,2,1,1,1] => 87 [3,1,1,1,1,1] => 255 [2,2,2,2] => 164 [2,2,2,1,1] => 164 [2,2,1,1,1,1] => 352 [2,1,1,1,1,1,1] => 1080 [1,1,1,1,1,1,1,1] => 4140 [9] => 3 [8,1] => 5 [7,2] => 5 [7,1,1] => 7 [6,3] => 12 [6,2,1] => 18 [6,1,1,1] => 30 [5,4] => 7 [5,3,1] => 10 [5,2,2] => 19 [5,2,1,1] => 27 [5,1,1,1,1] => 67 [4,4,1] => 23 [4,3,2] => 24 [4,3,1,1] => 34 [4,2,2,1] => 71 [4,2,1,1,1] => 117 [4,1,1,1,1,1] => 307 [3,3,3] => 42 [3,3,2,1] => 46 [3,3,1,1,1] => 102 [3,2,2,2] => 90 [3,2,2,1,1] => 128 [3,2,1,1,1,1] => 322 [3,1,1,1,1,1,1] => 1080 [2,2,2,2,1] => 339 [2,2,2,1,1,1] => 549 [2,2,1,1,1,1,1] => 1439 [2,1,1,1,1,1,1,1] => 5017 [1,1,1,1,1,1,1,1,1] => 21147 [10] => 4 [9,1] => 4 [8,2] => 11 [8,1,1] => 11 [7,3] => 5 [7,2,1] => 10 [7,1,1,1] => 20 [6,4] => 15 [6,3,1] => 19 [6,2,2] => 45 [6,2,1,1] => 45 [6,1,1,1,1] => 97 [5,5] => 10 [5,4,1] => 13 [5,3,2] => 15 [5,3,1,1] => 27 [5,2,2,1] => 43 [5,2,1,1,1] => 87 [5,1,1,1,1,1] => 255 [4,4,2] => 55 [4,4,1,1] => 55 [4,3,3] => 29 [4,3,2,1] => 53 [4,3,1,1,1] => 107 [4,2,2,2] => 195 [4,2,2,1,1] => 195 [4,2,1,1,1,1] => 419 [4,1,1,1,1,1,1] => 1283 [3,3,3,1] => 73 [3,3,2,2] => 83 [3,3,2,1,1] => 135 [3,3,1,1,1,1] => 367 [3,2,2,2,1] => 223 [3,2,2,1,1,1] => 449 [3,2,1,1,1,1,1] => 1335 [3,1,1,1,1,1,1,1] => 5017 [2,2,2,2,2] => 999 [2,2,2,2,1,1] => 999 [2,2,2,1,1,1,1] => 2119 [2,2,1,1,1,1,1,1] => 6503 [2,1,1,1,1,1,1,1,1] => 25287 [1,1,1,1,1,1,1,1,1,1] => 115975 [11] => 2 [10,1] => 5 [9,2] => 7 [9,1,1] => 9 [8,3] => 9 [8,2,1] => 18 [8,1,1,1] => 30 [7,4] => 7 [7,3,1] => 10 [7,2,2] => 19 [7,2,1,1] => 27 [7,1,1,1,1] => 67 [6,5] => 9 [6,4,1] => 23 [6,3,2] => 35 [6,3,1,1] => 47 [6,2,2,1] => 83 [6,2,1,1,1] => 137 [6,1,1,1,1,1] => 359 [5,5,1] => 15 [5,4,2] => 24 [5,4,1,1] => 34 [5,3,3] => 21 [5,3,2,1] => 37 [5,3,1,1,1] => 87 [5,2,2,2] => 90 [5,2,2,1,1] => 128 [5,2,1,1,1,1] => 322 [5,1,1,1,1,1,1] => 1080 [4,4,3] => 39 [4,4,2,1] => 98 [4,4,1,1,1] => 162 [4,3,3,1] => 59 [4,3,2,2] => 109 [4,3,2,1,1] => 155 [4,3,1,1,1,1] => 389 [4,2,2,2,1] => 398 [4,2,2,1,1,1] => 646 [4,2,1,1,1,1,1] => 1694 [4,1,1,1,1,1,1,1] => 5894 [3,3,3,2] => 115 [3,3,3,1,1] => 195 [3,3,2,2,1] => 207 [3,3,2,1,1,1] => 469 [3,3,1,1,1,1,1] => 1491 [3,2,2,2,2] => 503 [3,2,2,2,1,1] => 713 [3,2,2,1,1,1,1] => 1791 [3,2,1,1,1,1,1,1] => 6097 [3,1,1,1,1,1,1,1,1] => 25287 [2,2,2,2,2,1] => 2210 [2,2,2,2,1,1,1] => 3530 [2,2,2,1,1,1,1,1] => 9170 [2,2,1,1,1,1,1,1,1] => 32058 [2,1,1,1,1,1,1,1,1,1] => 137122 [1,1,1,1,1,1,1,1,1,1,1] => 678570 [12] => 6 [11,1] => 3 [10,2] => 11 [10,1,1] => 11 [9,3] => 10 [9,2,1] => 13 [9,1,1,1] => 25 [8,4] => 19 [8,3,1] => 16 [8,2,2] => 45 [8,2,1,1] => 45 [8,1,1,1,1] => 97 [7,5] => 5 [7,4,1] => 13 [7,3,2] => 15 [7,3,1,1] => 27 [7,2,2,1] => 43 [7,2,1,1,1] => 87 [7,1,1,1,1,1] => 255 [6,6] => 28 [6,5,1] => 16 [6,4,2] => 56 [6,4,1,1] => 56 [6,3,3] => 58 [6,3,2,1] => 72 [6,3,1,1,1] => 142 [6,2,2,2] => 226 [6,2,2,1,1] => 226 [6,2,1,1,1,1] => 486 [6,1,1,1,1,1,1] => 1486 [5,5,2] => 25 [5,5,1,1] => 37 [5,4,3] => 20 [5,4,2,1] => 53 [5,4,1,1,1] => 107 [5,3,3,1] => 46 [5,3,2,2] => 62 [5,3,2,1,1] => 114 [5,3,1,1,1,1] => 322 [5,2,2,2,1] => 223 [5,2,2,1,1,1] => 449 [5,2,1,1,1,1,1] => 1335 [5,1,1,1,1,1,1,1] => 5017 [4,4,4] => 111 [4,4,3,1] => 78 [4,4,2,2] => 261 [4,4,2,1,1] => 261 [4,4,1,1,1,1] => 561 [4,3,3,2] => 104 [4,3,3,1,1] => 168 [4,3,2,2,1] => 266 [4,3,2,1,1,1] => 536 [4,3,1,1,1,1,1] => 1590 [4,2,2,2,2] => 1163 [4,2,2,2,1,1] => 1163 [4,2,2,1,1,1,1] => 2471 [4,2,1,1,1,1,1,1] => 7583 [4,1,1,1,1,1,1,1,1] => 29427 [3,3,3,3] => 268 [3,3,3,2,1] => 268 [3,3,3,1,1,1] => 634 [3,3,2,2,2] => 406 [3,3,2,2,1,1] => 670 [3,3,2,1,1,1,1] => 1858 [3,3,1,1,1,1,1,1] => 6706 [3,2,2,2,2,1] => 1338 [3,2,2,2,1,1,1] => 2668 [3,2,2,1,1,1,1,1] => 7942 [3,2,1,1,1,1,1,1,1] => 30304 [3,1,1,1,1,1,1,1,1,1] => 137122 [2,2,2,2,2,2] => 6841 [2,2,2,2,2,1,1] => 6841 [2,2,2,2,1,1,1,1] => 14325 [2,2,2,1,1,1,1,1,1] => 43693 [2,2,1,1,1,1,1,1,1,1] => 170689 [2,1,1,1,1,1,1,1,1,1,1] => 794545 [1,1,1,1,1,1,1,1,1,1,1,1] => 4213597 ----------------------------------------------------------------------------- Created: May 26, 2016 at 21:32 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 26, 2016 at 21:32 by Martin Rubey