***************************************************************************** * 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: St000480 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$. ----------------------------------------------------------------------------- References: [1] Brylawski, T. The lattice of integer partitions [[MathSciNet:0325405]] ----------------------------------------------------------------------------- Code: @cached_function def P(k): return posets.IntegerPartitionsDominanceOrder(k) def statistic(pi): Q = P(pi.size()) return len(Q.lower_covers(Q(pi))) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [2] => 1 [1,1] => 0 [3] => 1 [2,1] => 1 [1,1,1] => 0 [4] => 1 [3,1] => 1 [2,2] => 1 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 1 [4,1] => 1 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 1 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 1 [5,1] => 1 [4,2] => 2 [4,1,1] => 1 [3,3] => 1 [3,2,1] => 2 [3,1,1,1] => 1 [2,2,2] => 1 [2,2,1,1] => 1 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 1 [6,1] => 1 [5,2] => 2 [5,1,1] => 1 [4,3] => 1 [4,2,1] => 2 [4,1,1,1] => 1 [3,3,1] => 1 [3,2,2] => 1 [3,2,1,1] => 2 [3,1,1,1,1] => 1 [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] => 0 [8] => 1 [7,1] => 1 [6,2] => 2 [6,1,1] => 1 [5,3] => 2 [5,2,1] => 2 [5,1,1,1] => 1 [4,4] => 1 [4,3,1] => 1 [4,2,2] => 2 [4,2,1,1] => 2 [4,1,1,1,1] => 1 [3,3,2] => 1 [3,3,1,1] => 1 [3,2,2,1] => 2 [3,2,1,1,1] => 2 [3,1,1,1,1,1] => 1 [2,2,2,2] => 1 [2,2,2,1,1] => 1 [2,2,1,1,1,1] => 1 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 0 [9] => 1 [8,1] => 1 [7,2] => 2 [7,1,1] => 1 [6,3] => 2 [6,2,1] => 2 [6,1,1,1] => 1 [5,4] => 1 [5,3,1] => 2 [5,2,2] => 2 [5,2,1,1] => 2 [5,1,1,1,1] => 1 [4,4,1] => 1 [4,3,2] => 2 [4,3,1,1] => 1 [4,2,2,1] => 2 [4,2,1,1,1] => 2 [4,1,1,1,1,1] => 1 [3,3,3] => 1 [3,3,2,1] => 2 [3,3,1,1,1] => 1 [3,2,2,2] => 1 [3,2,2,1,1] => 2 [3,2,1,1,1,1] => 2 [3,1,1,1,1,1,1] => 1 [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] => 0 [10] => 1 [9,1] => 1 [8,2] => 2 [8,1,1] => 1 [7,3] => 2 [7,2,1] => 2 [7,1,1,1] => 1 [6,4] => 2 [6,3,1] => 2 [6,2,2] => 2 [6,2,1,1] => 2 [6,1,1,1,1] => 1 [5,5] => 1 [5,4,1] => 1 [5,3,2] => 2 [5,3,1,1] => 2 [5,2,2,1] => 2 [5,2,1,1,1] => 2 [5,1,1,1,1,1] => 1 [4,4,2] => 2 [4,4,1,1] => 1 [4,3,3] => 1 [4,3,2,1] => 3 [4,3,1,1,1] => 1 [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] => 1 [3,3,3,1] => 1 [3,3,2,2] => 1 [3,3,2,1,1] => 2 [3,3,1,1,1,1] => 1 [3,2,2,2,1] => 2 [3,2,2,1,1,1] => 2 [3,2,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1] => 1 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 1 [2,2,2,1,1,1,1] => 1 [2,2,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 1 [10,1] => 1 [9,2] => 2 [9,1,1] => 1 [8,3] => 2 [8,2,1] => 2 [8,1,1,1] => 1 [7,4] => 2 [7,3,1] => 2 [7,2,2] => 2 [7,2,1,1] => 2 [7,1,1,1,1] => 1 [6,5] => 1 [6,4,1] => 2 [6,3,2] => 2 [6,3,1,1] => 2 [6,2,2,1] => 2 [6,2,1,1,1] => 2 [6,1,1,1,1,1] => 1 [5,5,1] => 1 [5,4,2] => 2 [5,4,1,1] => 1 [5,3,3] => 2 [5,3,2,1] => 3 [5,3,1,1,1] => 2 [5,2,2,2] => 2 [5,2,2,1,1] => 2 [5,2,1,1,1,1] => 2 [5,1,1,1,1,1,1] => 1 [4,4,3] => 1 [4,4,2,1] => 2 [4,4,1,1,1] => 1 [4,3,3,1] => 1 [4,3,2,2] => 2 [4,3,2,1,1] => 3 [4,3,1,1,1,1] => 1 [4,2,2,2,1] => 2 [4,2,2,1,1,1] => 2 [4,2,1,1,1,1,1] => 2 [4,1,1,1,1,1,1,1] => 1 [3,3,3,2] => 1 [3,3,3,1,1] => 1 [3,3,2,2,1] => 2 [3,3,2,1,1,1] => 2 [3,3,1,1,1,1,1] => 1 [3,2,2,2,2] => 1 [3,2,2,2,1,1] => 2 [3,2,2,1,1,1,1] => 2 [3,2,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1] => 1 [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] => 0 [12] => 1 [11,1] => 1 [10,2] => 2 [10,1,1] => 1 [9,3] => 2 [9,2,1] => 2 [9,1,1,1] => 1 [8,4] => 2 [8,3,1] => 2 [8,2,2] => 2 [8,2,1,1] => 2 [8,1,1,1,1] => 1 [7,5] => 2 [7,4,1] => 2 [7,3,2] => 2 [7,3,1,1] => 2 [7,2,2,1] => 2 [7,2,1,1,1] => 2 [7,1,1,1,1,1] => 1 [6,6] => 1 [6,5,1] => 1 [6,4,2] => 3 [6,4,1,1] => 2 [6,3,3] => 2 [6,3,2,1] => 3 [6,3,1,1,1] => 2 [6,2,2,2] => 2 [6,2,2,1,1] => 2 [6,2,1,1,1,1] => 2 [6,1,1,1,1,1,1] => 1 [5,5,2] => 2 [5,5,1,1] => 1 [5,4,3] => 2 [5,4,2,1] => 2 [5,4,1,1,1] => 1 [5,3,3,1] => 2 [5,3,2,2] => 2 [5,3,2,1,1] => 3 [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] => 1 [4,4,4] => 1 [4,4,3,1] => 1 [4,4,2,2] => 2 [4,4,2,1,1] => 2 [4,4,1,1,1,1] => 1 [4,3,3,2] => 2 [4,3,3,1,1] => 1 [4,3,2,2,1] => 3 [4,3,2,1,1,1] => 3 [4,3,1,1,1,1,1] => 1 [4,2,2,2,2] => 2 [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] => 1 [3,3,3,3] => 1 [3,3,3,2,1] => 2 [3,3,3,1,1,1] => 1 [3,3,2,2,2] => 1 [3,3,2,2,1,1] => 2 [3,3,2,1,1,1,1] => 2 [3,3,1,1,1,1,1,1] => 1 [3,2,2,2,2,1] => 2 [3,2,2,2,1,1,1] => 2 [3,2,2,1,1,1,1,1] => 2 [3,2,1,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 1 [2,2,2,2,1,1,1,1] => 1 [2,2,2,1,1,1,1,1,1] => 1 [2,2,1,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: May 09, 2016 at 09:22 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 21:36 by Martin Rubey