***************************************************************************** * 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: St000212 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. Summing over all partitions of $n$ yields the sequence $$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$ which is [[oeis:A237770]]. The references in this sequence of the OEIS indicate a connection with Baxter permutations. ----------------------------------------------------------------------------- References: [1] Dulucq, S., Guibert, O. Stack words, standard tableaux and Baxter permutations [[MathSciNet:1417289]] [[DOI:10.1016/s0012-365x(96)83009-3]] ----------------------------------------------------------------------------- Code: def statistic(x): l = [any(any(f == e+1 for (e,f) in zip(row, row[1:])) for row in T) for T in StandardTableaux(x)] return l.count(False) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 1 [1,1,1] => 1 [4] => 0 [3,1] => 0 [2,2] => 1 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 0 [4,1] => 0 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 3 [2,1,1,1] => 3 [1,1,1,1,1] => 1 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 0 [3,3] => 1 [3,2,1] => 5 [3,1,1,1] => 3 [2,2,2] => 2 [2,2,1,1] => 6 [2,1,1,1,1] => 4 [1,1,1,1,1,1] => 1 [7] => 0 [6,1] => 0 [5,2] => 0 [5,1,1] => 0 [4,3] => 1 [4,2,1] => 2 [4,1,1,1] => 1 [3,3,1] => 5 [3,2,2] => 6 [3,2,1,1] => 14 [3,1,1,1,1] => 6 [2,2,2,1] => 8 [2,2,1,1,1] => 10 [2,1,1,1,1,1] => 5 [1,1,1,1,1,1,1] => 1 [8] => 0 [7,1] => 0 [6,2] => 0 [6,1,1] => 0 [5,3] => 0 [5,2,1] => 0 [5,1,1,1] => 0 [4,4] => 1 [4,3,1] => 8 [4,2,2] => 6 [4,2,1,1] => 11 [4,1,1,1,1] => 4 [3,3,2] => 10 [3,3,1,1] => 16 [3,2,2,1] => 28 [3,2,1,1,1] => 30 [3,1,1,1,1,1] => 10 [2,2,2,2] => 6 [2,2,2,1,1] => 18 [2,2,1,1,1,1] => 15 [2,1,1,1,1,1,1] => 6 [1,1,1,1,1,1,1,1] => 1 [9] => 0 [8,1] => 0 [7,2] => 0 [7,1,1] => 0 [6,3] => 0 [6,2,1] => 0 [6,1,1,1] => 0 [5,4] => 1 [5,3,1] => 3 [5,2,2] => 2 [5,2,1,1] => 3 [5,1,1,1,1] => 1 [4,4,1] => 7 [4,3,2] => 23 [4,3,1,1] => 34 [4,2,2,1] => 37 [4,2,1,1,1] => 35 [4,1,1,1,1,1] => 10 [3,3,3] => 6 [3,3,2,1] => 54 [3,3,1,1,1] => 40 [3,2,2,2] => 28 [3,2,2,1,1] => 76 [3,2,1,1,1,1] => 55 [3,1,1,1,1,1,1] => 15 [2,2,2,2,1] => 24 [2,2,2,1,1,1] => 33 [2,2,1,1,1,1,1] => 21 [2,1,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1,1] => 1 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 0 [7,3] => 0 [7,2,1] => 0 [7,1,1,1] => 0 [6,4] => 0 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 1 [5,4,1] => 11 [5,3,2] => 18 [5,3,1,1] => 24 [5,2,2,1] => 22 [5,2,1,1,1] => 19 [5,1,1,1,1,1] => 5 [4,4,2] => 23 [4,4,1,1] => 31 [4,3,3] => 22 [4,3,2,1] => 148 [4,3,1,1,1] => 105 [4,2,2,2] => 53 [4,2,2,1,1] => 130 [4,2,1,1,1,1] => 85 [4,1,1,1,1,1,1] => 20 [3,3,3,1] => 44 [3,3,2,2] => 72 [3,3,2,1,1] => 170 [3,3,1,1,1,1] => 85 [3,2,2,2,1] => 128 [3,2,2,1,1,1] => 164 [3,2,1,1,1,1,1] => 91 [3,1,1,1,1,1,1,1] => 21 [2,2,2,2,2] => 18 [2,2,2,2,1,1] => 57 [2,2,2,1,1,1,1] => 54 [2,2,1,1,1,1,1,1] => 28 [2,1,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 0 [10,1] => 0 [9,2] => 0 [9,1,1] => 0 [8,3] => 0 [8,2,1] => 0 [8,1,1,1] => 0 [7,4] => 0 [7,3,1] => 0 [7,2,2] => 0 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 1 [6,4,1] => 4 [6,3,2] => 5 [6,3,1,1] => 6 [6,2,2,1] => 5 [6,2,1,1,1] => 4 [6,1,1,1,1,1] => 1 [5,5,1] => 9 [5,4,2] => 49 [5,4,1,1] => 63 [5,3,3] => 31 [5,3,2,1] => 158 [5,3,1,1,1] => 107 [5,2,2,2] => 51 [5,2,2,1,1] => 113 [5,2,1,1,1,1] => 69 [5,1,1,1,1,1,1] => 15 [4,4,3] => 39 [4,4,2,1] => 165 [4,4,1,1,1] => 105 [4,3,3,1] => 180 [4,3,2,2] => 250 [4,3,2,1,1] => 553 [4,3,1,1,1,1] => 265 [4,2,2,2,1] => 287 [4,2,2,1,1,1] => 346 [4,2,1,1,1,1,1] => 175 [4,1,1,1,1,1,1,1] => 35 [3,3,3,2] => 110 [3,3,3,1,1] => 174 [3,3,2,2,1] => 370 [3,3,2,1,1,1] => 419 [3,3,1,1,1,1,1] => 161 [3,2,2,2,2] => 118 [3,2,2,2,1,1] => 349 [3,2,2,1,1,1,1] => 309 [3,2,1,1,1,1,1,1] => 140 [3,1,1,1,1,1,1,1,1] => 28 [2,2,2,2,2,1] => 75 [2,2,2,2,1,1,1] => 111 [2,2,2,1,1,1,1,1] => 82 [2,2,1,1,1,1,1,1,1] => 36 [2,1,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 0 [11,1] => 0 [10,2] => 0 [10,1,1] => 0 [9,3] => 0 [9,2,1] => 0 [9,1,1,1] => 0 [8,4] => 0 [8,3,1] => 0 [8,2,2] => 0 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 0 [7,4,1] => 0 [7,3,2] => 0 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 1 [6,5,1] => 14 [6,4,2] => 35 [6,4,1,1] => 42 [6,3,3] => 20 [6,3,2,1] => 80 [6,3,1,1,1] => 52 [6,2,2,2] => 25 [6,2,2,1,1] => 50 [6,2,1,1,1,1] => 29 [6,1,1,1,1,1,1] => 6 [5,5,2] => 41 [5,5,1,1] => 51 [5,4,3] => 109 [5,4,2,1] => 413 [5,4,1,1,1] => 257 [5,3,3,1] => 301 [5,3,2,2] => 369 [5,3,2,1,1] => 761 [5,3,1,1,1,1] => 351 [5,2,2,2,1] => 347 [5,2,2,1,1,1] => 397 [5,2,1,1,1,1,1] => 189 [5,1,1,1,1,1,1,1] => 35 [4,4,4] => 22 [4,4,3,1] => 353 [4,4,2,2] => 339 [4,4,2,1,1] => 693 [4,4,1,1,1,1] => 295 [4,3,3,2] => 518 [4,3,3,1,1] => 802 [4,3,2,2,1] => 1460 [4,3,2,1,1,1] => 1583 [4,3,1,1,1,1,1] => 581 [4,2,2,2,2] => 334 [4,2,2,2,1,1] => 925 [4,2,2,1,1,1,1] => 776 [4,2,1,1,1,1,1,1] => 322 [4,1,1,1,1,1,1,1,1] => 56 [3,3,3,3] => 72 [3,3,3,2,1] => 654 [3,3,3,1,1,1] => 508 [3,3,2,2,2] => 416 [3,3,2,2,1,1] => 1138 [3,3,2,1,1,1,1] => 889 [3,3,1,1,1,1,1,1] => 280 [3,2,2,2,2,1] => 542 [3,2,2,2,1,1,1] => 769 [3,2,2,1,1,1,1,1] => 531 [3,2,1,1,1,1,1,1,1] => 204 [3,1,1,1,1,1,1,1,1,1] => 36 [2,2,2,2,2,2] => 57 [2,2,2,2,2,1,1] => 186 [2,2,2,2,1,1,1,1] => 193 [2,2,2,1,1,1,1,1,1] => 118 [2,2,1,1,1,1,1,1,1,1] => 45 [2,1,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Aug 21, 2014 at 14:22 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 16:35 by Martin Rubey