***************************************************************************** * 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: St001055 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses. ----------------------------------------------------------------------------- References: [1] Row, A. Ordered Nim game [[MathOverflow:286925]] ----------------------------------------------------------------------------- Code: @cached_function def statistic(la): """Return the Grundy value of the partition for the game where one may remove a positive number of cells in one row. """ def children(la): if len(la) == 0: return for i in range(len(la)-1): for j in range(1, la[i]-la[i+1]+1): mu = [la[k] if k != i else la[k]-j for k in range(len(la))] yield Partition(mu) for j in range(1, la[-1]+1): mu = la[:-1] + [la[-1]-j] yield Partition(mu) l = [statistic(mu) for mu in children(la)] i = 0 while i in l: i += 1 return i ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 2 [1,1] => 0 [3] => 3 [2,1] => 1 [1,1,1] => 1 [4] => 4 [3,1] => 2 [2,2] => 0 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 5 [4,1] => 3 [3,2] => 1 [3,1,1] => 3 [2,2,1] => 1 [2,1,1,1] => 1 [1,1,1,1,1] => 1 [6] => 6 [5,1] => 4 [4,2] => 2 [4,1,1] => 2 [3,3] => 0 [3,2,1] => 0 [3,1,1,1] => 2 [2,2,2] => 2 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 7 [6,1] => 5 [5,2] => 3 [5,1,1] => 5 [4,3] => 1 [4,2,1] => 3 [4,1,1,1] => 3 [3,3,1] => 1 [3,2,2] => 3 [3,2,1,1] => 1 [3,1,1,1,1] => 3 [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] => 8 [7,1] => 6 [6,2] => 4 [6,1,1] => 4 [5,3] => 2 [5,2,1] => 2 [5,1,1,1] => 4 [4,4] => 0 [4,3,1] => 0 [4,2,2] => 0 [4,2,1,1] => 2 [4,1,1,1,1] => 2 [3,3,2] => 2 [3,3,1,1] => 0 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 2 [2,2,2,2] => 0 [2,2,2,1,1] => 0 [2,2,1,1,1,1] => 0 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => 0 [9] => 9 [8,1] => 7 [7,2] => 5 [7,1,1] => 7 [6,3] => 3 [6,2,1] => 5 [6,1,1,1] => 5 [5,4] => 1 [5,3,1] => 3 [5,2,2] => 1 [5,2,1,1] => 3 [5,1,1,1,1] => 5 [4,4,1] => 1 [4,3,2] => 3 [4,3,1,1] => 1 [4,2,2,1] => 3 [4,2,1,1,1] => 3 [4,1,1,1,1,1] => 3 [3,3,3] => 3 [3,3,2,1] => 1 [3,3,1,1,1] => 1 [3,2,2,2] => 1 [3,2,2,1,1] => 1 [3,2,1,1,1,1] => 1 [3,1,1,1,1,1,1] => 3 [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] => 10 [9,1] => 8 [8,2] => 6 [8,1,1] => 6 [7,3] => 4 [7,2,1] => 4 [7,1,1,1] => 6 [6,4] => 2 [6,3,1] => 2 [6,2,2] => 6 [6,2,1,1] => 4 [6,1,1,1,1] => 4 [5,5] => 0 [5,4,1] => 0 [5,3,2] => 0 [5,3,1,1] => 2 [5,2,2,1] => 2 [5,2,1,1,1] => 2 [5,1,1,1,1,1] => 4 [4,4,2] => 2 [4,4,1,1] => 0 [4,3,3] => 2 [4,3,2,1] => 0 [4,3,1,1,1] => 0 [4,2,2,2] => 2 [4,2,2,1,1] => 2 [4,2,1,1,1,1] => 2 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 2 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 0 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 2 [2,2,2,2,2] => 2 [2,2,2,2,1,1] => 0 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 11 [10,1] => 9 [9,2] => 7 [9,1,1] => 9 [8,3] => 5 [8,2,1] => 7 [8,1,1,1] => 7 [7,4] => 3 [7,3,1] => 5 [7,2,2] => 7 [7,2,1,1] => 5 [7,1,1,1,1] => 7 [6,5] => 1 [6,4,1] => 3 [6,3,2] => 1 [6,3,1,1] => 3 [6,2,2,1] => 5 [6,2,1,1,1] => 5 [6,1,1,1,1,1] => 5 [5,5,1] => 1 [5,4,2] => 3 [5,4,1,1] => 1 [5,3,3] => 1 [5,3,2,1] => 3 [5,3,1,1,1] => 3 [5,2,2,2] => 3 [5,2,2,1,1] => 3 [5,2,1,1,1,1] => 3 [5,1,1,1,1,1,1] => 5 [4,4,3] => 3 [4,4,2,1] => 1 [4,4,1,1,1] => 1 [4,3,3,1] => 3 [4,3,2,2] => 1 [4,3,2,1,1] => 1 [4,3,1,1,1,1] => 1 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => 3 [4,2,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1] => 3 [3,3,3,2] => 1 [3,3,3,1,1] => 3 [3,3,2,2,1] => 1 [3,3,2,1,1,1] => 1 [3,3,1,1,1,1,1] => 1 [3,2,2,2,2] => 3 [3,2,2,2,1,1] => 1 [3,2,2,1,1,1,1] => 1 [3,2,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1] => 3 [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] => 12 [11,1] => 10 [10,2] => 8 [10,1,1] => 8 [9,3] => 6 [9,2,1] => 6 [9,1,1,1] => 8 [8,4] => 4 [8,3,1] => 4 [8,2,2] => 4 [8,2,1,1] => 6 [8,1,1,1,1] => 6 [7,5] => 2 [7,4,1] => 2 [7,3,2] => 6 [7,3,1,1] => 4 [7,2,2,1] => 4 [7,2,1,1,1] => 4 [7,1,1,1,1,1] => 6 [6,6] => 0 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 2 [6,3,3] => 0 [6,3,2,1] => 2 [6,3,1,1,1] => 2 [6,2,2,2] => 4 [6,2,2,1,1] => 4 [6,2,1,1,1,1] => 4 [6,1,1,1,1,1,1] => 4 [5,5,2] => 2 [5,5,1,1] => 0 [5,4,3] => 2 [5,4,2,1] => 0 [5,4,1,1,1] => 0 [5,3,3,1] => 0 [5,3,2,2] => 2 [5,3,2,1,1] => 2 [5,3,1,1,1,1] => 2 [5,2,2,2,1] => 2 [5,2,2,1,1,1] => 2 [5,2,1,1,1,1,1] => 2 [5,1,1,1,1,1,1,1] => 4 [4,4,4] => 4 [4,4,3,1] => 2 [4,4,2,2] => 0 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 0 [4,3,3,1,1] => 2 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 2 [4,2,2,1,1,1,1] => 2 [4,2,1,1,1,1,1,1] => 2 [4,1,1,1,1,1,1,1,1] => 2 [3,3,3,3] => 0 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 2 [3,3,2,2,2] => 2 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 0 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2] => 0 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Nov 25, 2017 at 13:45 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 25, 2017 at 13:45 by Martin Rubey