***************************************************************************** * 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: St000145 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$ ----------------------------------------------------------------------------- References: [1] Dyson, F. J. Some guesses in the theory of partitions [[MathSciNet:3077150]] ----------------------------------------------------------------------------- Code: def statistic(L): return L[0] - len(L) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 1 [1,1] => -1 [3] => 2 [2,1] => 0 [1,1,1] => -2 [4] => 3 [3,1] => 1 [2,2] => 0 [2,1,1] => -1 [1,1,1,1] => -3 [5] => 4 [4,1] => 2 [3,2] => 1 [3,1,1] => 0 [2,2,1] => -1 [2,1,1,1] => -2 [1,1,1,1,1] => -4 [6] => 5 [5,1] => 3 [4,2] => 2 [4,1,1] => 1 [3,3] => 1 [3,2,1] => 0 [3,1,1,1] => -1 [2,2,2] => -1 [2,2,1,1] => -2 [2,1,1,1,1] => -3 [1,1,1,1,1,1] => -5 [7] => 6 [6,1] => 4 [5,2] => 3 [5,1,1] => 2 [4,3] => 2 [4,2,1] => 1 [4,1,1,1] => 0 [3,3,1] => 0 [3,2,2] => 0 [3,2,1,1] => -1 [3,1,1,1,1] => -2 [2,2,2,1] => -2 [2,2,1,1,1] => -3 [2,1,1,1,1,1] => -4 [1,1,1,1,1,1,1] => -6 [8] => 7 [7,1] => 5 [6,2] => 4 [6,1,1] => 3 [5,3] => 3 [5,2,1] => 2 [5,1,1,1] => 1 [4,4] => 2 [4,3,1] => 1 [4,2,2] => 1 [4,2,1,1] => 0 [4,1,1,1,1] => -1 [3,3,2] => 0 [3,3,1,1] => -1 [3,2,2,1] => -1 [3,2,1,1,1] => -2 [3,1,1,1,1,1] => -3 [2,2,2,2] => -2 [2,2,2,1,1] => -3 [2,2,1,1,1,1] => -4 [2,1,1,1,1,1,1] => -5 [1,1,1,1,1,1,1,1] => -7 [9] => 8 [8,1] => 6 [7,2] => 5 [7,1,1] => 4 [6,3] => 4 [6,2,1] => 3 [6,1,1,1] => 2 [5,4] => 3 [5,3,1] => 2 [5,2,2] => 2 [5,2,1,1] => 1 [5,1,1,1,1] => 0 [4,4,1] => 1 [4,3,2] => 1 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => -1 [4,1,1,1,1,1] => -2 [3,3,3] => 0 [3,3,2,1] => -1 [3,3,1,1,1] => -2 [3,2,2,2] => -1 [3,2,2,1,1] => -2 [3,2,1,1,1,1] => -3 [3,1,1,1,1,1,1] => -4 [2,2,2,2,1] => -3 [2,2,2,1,1,1] => -4 [2,2,1,1,1,1,1] => -5 [2,1,1,1,1,1,1,1] => -6 [1,1,1,1,1,1,1,1,1] => -8 [10] => 9 [9,1] => 7 [8,2] => 6 [8,1,1] => 5 [7,3] => 5 [7,2,1] => 4 [7,1,1,1] => 3 [6,4] => 4 [6,3,1] => 3 [6,2,2] => 3 [6,2,1,1] => 2 [6,1,1,1,1] => 1 [5,5] => 3 [5,4,1] => 2 [5,3,2] => 2 [5,3,1,1] => 1 [5,2,2,1] => 1 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => -1 [4,4,2] => 1 [4,4,1,1] => 0 [4,3,3] => 1 [4,3,2,1] => 0 [4,3,1,1,1] => -1 [4,2,2,2] => 0 [4,2,2,1,1] => -1 [4,2,1,1,1,1] => -2 [4,1,1,1,1,1,1] => -3 [3,3,3,1] => -1 [3,3,2,2] => -1 [3,3,2,1,1] => -2 [3,3,1,1,1,1] => -3 [3,2,2,2,1] => -2 [3,2,2,1,1,1] => -3 [3,2,1,1,1,1,1] => -4 [3,1,1,1,1,1,1,1] => -5 [2,2,2,2,2] => -3 [2,2,2,2,1,1] => -4 [2,2,2,1,1,1,1] => -5 [2,2,1,1,1,1,1,1] => -6 [2,1,1,1,1,1,1,1,1] => -7 [1,1,1,1,1,1,1,1,1,1] => -9 ----------------------------------------------------------------------------- Created: Jul 03, 2013 at 14:34 by Olivier Mallet ----------------------------------------------------------------------------- Last Updated: May 29, 2015 at 16:57 by Martin Rubey