***************************************************************************** * 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: St000506 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def tableau_ascents(t): r""" The (sorted list) of ascents of the standard tableau `t`. An *ascent* of a standard tableau `t` is an entry `i` such that `i+1` apears to the right or above `i` in `t` (in English notation for tableaux). """ locations = {} for (i, row) in enumerate(t): for (j, entry) in enumerate(row): locations[entry] = (i, j) ascents = [t.size()] for i in range(1, t.size()): # ascent means i+1 appears to the right or above x, _ = locations[i] u, _ = locations[i+1] if u <= x: ascents.append(i) return sorted(ascents) def is_desarrangement_tableau(t): r""" Test whether a tableau is a desarrangement tableau. A *desarrangement tableau* is a standard tableau whose first ascent is even. """ return min(tableau_ascents(Tableau(t))) % 2 == 0 def statistic(la): return len([t for t in StandardTableaux(la) if is_desarrangement_tableau(t)]) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 0 [3,1] => 1 [2,2] => 1 [2,1,1] => 1 [1,1,1,1] => 1 [5] => 0 [4,1] => 1 [3,2] => 2 [3,1,1] => 2 [2,2,1] => 2 [2,1,1,1] => 2 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 1 [4,2] => 3 [4,1,1] => 3 [3,3] => 2 [3,2,1] => 6 [3,1,1,1] => 4 [2,2,2] => 2 [2,2,1,1] => 4 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 0 [6,1] => 1 [5,2] => 4 [5,1,1] => 4 [4,3] => 5 [4,2,1] => 12 [4,1,1,1] => 7 [3,3,1] => 8 [3,2,2] => 8 [3,2,1,1] => 14 [3,1,1,1,1] => 6 [2,2,2,1] => 6 [2,2,1,1,1] => 6 [2,1,1,1,1,1] => 3 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 1 [6,2] => 5 [6,1,1] => 5 [5,3] => 9 [5,2,1] => 20 [5,1,1,1] => 11 [4,4] => 5 [4,3,1] => 25 [4,2,2] => 20 [4,2,1,1] => 33 [4,1,1,1,1] => 13 [3,3,2] => 16 [3,3,1,1] => 22 [3,2,2,1] => 28 [3,2,1,1,1] => 26 [3,1,1,1,1,1] => 9 [2,2,2,2] => 6 [2,2,2,1,1] => 12 [2,2,1,1,1,1] => 9 [2,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,1,1] => 1 [9] => 0 [8,1] => 1 [7,2] => 6 [7,1,1] => 6 [6,3] => 14 [6,2,1] => 30 [6,1,1,1] => 16 [5,4] => 14 [5,3,1] => 54 [5,2,2] => 40 [5,2,1,1] => 64 [5,1,1,1,1] => 24 [4,4,1] => 30 [4,3,2] => 61 [4,3,1,1] => 80 [4,2,2,1] => 81 [4,2,1,1,1] => 72 [4,1,1,1,1,1] => 22 [3,3,3] => 16 [3,3,2,1] => 66 [3,3,1,1,1] => 48 [3,2,2,2] => 34 [3,2,2,1,1] => 66 [3,2,1,1,1,1] => 44 [3,1,1,1,1,1,1] => 12 [2,2,2,2,1] => 18 [2,2,2,1,1,1] => 21 [2,2,1,1,1,1,1] => 12 [2,1,1,1,1,1,1,1] => 4 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 1 [8,2] => 7 [8,1,1] => 7 [7,3] => 20 [7,2,1] => 42 [7,1,1,1] => 22 [6,4] => 28 [6,3,1] => 98 [6,2,2] => 70 [6,2,1,1] => 110 [6,1,1,1,1] => 40 [5,5] => 14 [5,4,1] => 98 [5,3,2] => 155 [5,3,1,1] => 198 [5,2,2,1] => 185 [5,2,1,1,1] => 160 [5,1,1,1,1,1] => 46 [4,4,2] => 91 [4,4,1,1] => 110 [4,3,3] => 77 [4,3,2,1] => 288 [4,3,1,1,1] => 200 [4,2,2,2] => 115 [4,2,2,1,1] => 219 [4,2,1,1,1,1] => 138 [4,1,1,1,1,1,1] => 34 [3,3,3,1] => 82 [3,3,2,2] => 100 [3,3,2,1,1] => 180 [3,3,1,1,1,1] => 92 [3,2,2,2,1] => 118 [3,2,2,1,1,1] => 131 [3,2,1,1,1,1,1] => 68 [3,1,1,1,1,1,1,1] => 16 [2,2,2,2,2] => 18 [2,2,2,2,1,1] => 39 [2,2,2,1,1,1,1] => 33 [2,2,1,1,1,1,1,1] => 16 [2,1,1,1,1,1,1,1,1] => 4 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 0 [10,1] => 1 [9,2] => 8 [9,1,1] => 8 [8,3] => 27 [8,2,1] => 56 [8,1,1,1] => 29 [7,4] => 48 [7,3,1] => 160 [7,2,2] => 112 [7,2,1,1] => 174 [7,1,1,1,1] => 62 [6,5] => 42 [6,4,1] => 224 [6,3,2] => 323 [6,3,1,1] => 406 [6,2,2,1] => 365 [6,2,1,1,1] => 310 [6,1,1,1,1,1] => 86 [5,5,1] => 112 [5,4,2] => 344 [5,4,1,1] => 406 [5,3,3] => 232 [5,3,2,1] => 826 [5,3,1,1,1] => 558 [5,2,2,2] => 300 [5,2,2,1,1] => 564 [5,2,1,1,1,1] => 344 [5,1,1,1,1,1,1] => 80 [4,4,3] => 168 [4,4,2,1] => 489 [4,4,1,1,1] => 310 [4,3,3,1] => 447 [4,3,2,2] => 503 [4,3,2,1,1] => 887 [4,3,1,1,1,1] => 430 [4,2,2,2,1] => 452 [4,2,2,1,1,1] => 488 [4,2,1,1,1,1,1] => 240 [4,1,1,1,1,1,1,1] => 50 [3,3,3,2] => 182 [3,3,3,1,1] => 262 [3,3,2,2,1] => 398 [3,3,2,1,1,1] => 403 [3,3,1,1,1,1,1] => 160 [3,2,2,2,2] => 136 [3,2,2,2,1,1] => 288 [3,2,2,1,1,1,1] => 232 [3,2,1,1,1,1,1,1] => 100 [3,1,1,1,1,1,1,1,1] => 20 [2,2,2,2,2,1] => 57 [2,2,2,2,1,1,1] => 72 [2,2,2,1,1,1,1,1] => 49 [2,2,1,1,1,1,1,1,1] => 20 [2,1,1,1,1,1,1,1,1,1] => 5 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 0 [11,1] => 1 [10,2] => 9 [10,1,1] => 9 [9,3] => 35 [9,2,1] => 72 [9,1,1,1] => 37 [8,4] => 75 [8,3,1] => 243 [8,2,2] => 168 [8,2,1,1] => 259 [8,1,1,1,1] => 91 [7,5] => 90 [7,4,1] => 432 [7,3,2] => 595 [7,3,1,1] => 740 [7,2,2,1] => 651 [7,2,1,1,1] => 546 [7,1,1,1,1,1] => 148 [6,6] => 42 [6,5,1] => 378 [6,4,2] => 891 [6,4,1,1] => 1036 [6,3,3] => 555 [6,3,2,1] => 1920 [6,3,1,1,1] => 1274 [6,2,2,2] => 665 [6,2,2,1,1] => 1239 [6,2,1,1,1,1] => 740 [6,1,1,1,1,1,1] => 166 [5,5,2] => 456 [5,5,1,1] => 518 [5,4,3] => 744 [5,4,2,1] => 2065 [5,4,1,1,1] => 1274 [5,3,3,1] => 1505 [5,3,2,2] => 1629 [5,3,2,1,1] => 2835 [5,3,1,1,1,1] => 1332 [5,2,2,2,1] => 1316 [5,2,2,1,1,1] => 1396 [5,2,1,1,1,1,1] => 664 [5,1,1,1,1,1,1,1] => 130 [4,4,4] => 168 [4,4,3,1] => 1104 [4,4,2,2] => 992 [4,4,2,1,1] => 1686 [4,4,1,1,1,1] => 740 [4,3,3,2] => 1132 [4,3,3,1,1] => 1596 [4,3,2,2,1] => 2240 [4,3,2,1,1,1] => 2208 [4,3,1,1,1,1,1] => 830 [4,2,2,2,2] => 588 [4,2,2,2,1,1] => 1228 [4,2,2,1,1,1,1] => 960 [4,2,1,1,1,1,1,1] => 390 [4,1,1,1,1,1,1,1,1] => 70 [3,3,3,3] => 182 [3,3,3,2,1] => 842 [3,3,3,1,1,1] => 665 [3,3,2,2,2] => 534 [3,3,2,2,1,1] => 1089 [3,3,2,1,1,1,1] => 795 [3,3,1,1,1,1,1,1] => 260 [3,2,2,2,2,1] => 481 [3,2,2,2,1,1,1] => 592 [3,2,2,1,1,1,1,1] => 381 [3,2,1,1,1,1,1,1,1] => 140 [3,1,1,1,1,1,1,1,1,1] => 25 [2,2,2,2,2,2] => 57 [2,2,2,2,2,1,1] => 129 [2,2,2,2,1,1,1,1] => 121 [2,2,2,1,1,1,1,1,1] => 69 [2,2,1,1,1,1,1,1,1,1] => 25 [2,1,1,1,1,1,1,1,1,1,1] => 5 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: May 24, 2016 at 23:10 by Franco Saliola ----------------------------------------------------------------------------- Last Updated: Jun 11, 2016 at 01:03 by Martin Rubey