***************************************************************************** * 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: St000949 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: Gives the number of generalised tilting modules of the corresponding LNakayama algebra. ----------------------------------------------------------------------------- References: [1] [[https://en.wikipedia.org/wiki/Tilting_theory]] ----------------------------------------------------------------------------- Code: gap('LoadPackage("QPA");') def NthRadical(M, n): if n == 0: f = gap.IdentityMapping(M) else: f = gap.RadicalOfModuleInclusion(M) N = gap.Source(f) for i in range(n-1): h = gap.RadicalOfModuleInclusion(N); N = gap.Source(h) f = h * f return f def ARQuiverNak(A): injA = gap.IndecInjectiveModules(A) L = [gap.Source(NthRadical(inj, j)) for inj in injA for j in range(gap.Dimension(inj).sage())] return L def kupisch(D): """ sage: [kupisch(D) for D in DyckWords(3)] [[2, 2, 2, 1], [2, 3, 2, 1], [3, 2, 2, 1], [3, 3, 2, 1], [4, 3, 2, 1]] sage: all(kupisch(D) == [a+2 for a in D.reverse().to_area_sequence()[::-1]] + [1] for D in DyckWords(5)) """ H = D.heights() return [1+H[i] for i, s in enumerate(D) if s == 0]+[1] def statistic(D): D = DyckWord(D) K = kupisch(D) A = gap.NakayamaAlgebra(K, gap.GF(3)) g = gap.GlobalDimensionOfAlgebra(A,30) L = ARQuiverNak(A) LL = [x for x in L if not gap.IsProjectiveModule(x) or not gap.IsInjectiveModule(x)] r = len(gap.SimpleModules(A)) - (len(L) - len(LL)) S = [[LL[i-1] for i in s] for s in Subsets(len(LL), r)] return sum(1 for x in S if gap.N_RigidModule(gap.DirectSumOfQPAModules(x) , g)) ----------------------------------------------------------------------------- Statistic values: [1,0] => 2 [1,0,1,0] => 3 [1,1,0,0] => 5 [1,0,1,0,1,0] => 4 [1,0,1,1,0,0] => 7 [1,1,0,0,1,0] => 7 [1,1,0,1,0,0] => 9 [1,1,1,0,0,0] => 14 [1,0,1,0,1,0,1,0] => 5 [1,0,1,0,1,1,0,0] => 9 [1,0,1,1,0,0,1,0] => 10 [1,0,1,1,0,1,0,0] => 11 [1,0,1,1,1,0,0,0] => 19 [1,1,0,0,1,0,1,0] => 9 [1,1,0,0,1,1,0,0] => 16 [1,1,0,1,0,0,1,0] => 11 [1,1,0,1,0,1,0,0] => 15 [1,1,0,1,1,0,0,0] => 23 [1,1,1,0,0,0,1,0] => 19 [1,1,1,0,0,1,0,0] => 23 [1,1,1,0,1,0,0,0] => 28 [1,1,1,1,0,0,0,0] => 42 [1,0,1,0,1,0,1,0,1,0] => 6 [1,0,1,0,1,0,1,1,0,0] => 11 [1,0,1,0,1,1,0,0,1,0] => 13 [1,0,1,0,1,1,0,1,0,0] => 13 [1,0,1,0,1,1,1,0,0,0] => 24 [1,0,1,1,0,0,1,0,1,0] => 13 [1,0,1,1,0,0,1,1,0,0] => 23 [1,0,1,1,0,1,0,0,1,0] => 13 [1,0,1,1,0,1,0,1,0,0] => 18 [1,0,1,1,0,1,1,0,0,0] => 27 [1,0,1,1,1,0,0,0,1,0] => 26 [1,0,1,1,1,0,0,1,0,0] => 32 [1,0,1,1,1,0,1,0,0,0] => 33 [1,0,1,1,1,1,0,0,0,0] => 56 [1,1,0,0,1,0,1,0,1,0] => 11 [1,1,0,0,1,0,1,1,0,0] => 20 [1,1,0,0,1,1,0,0,1,0] => 23 [1,1,0,0,1,1,0,1,0,0] => 24 [1,1,0,0,1,1,1,0,0,0] => 43 [1,1,0,1,0,0,1,0,1,0] => 13 [1,1,0,1,0,0,1,1,0,0] => 24 [1,1,0,1,0,1,0,0,1,0] => 18 [1,1,0,1,0,1,0,1,0,0] => 22 [1,1,0,1,0,1,1,0,0,0] => 37 [1,1,0,1,1,0,0,0,1,0] => 32 [1,1,0,1,1,0,0,1,0,0] => 32 [1,1,0,1,1,0,1,0,0,0] => 43 [1,1,0,1,1,1,0,0,0,0] => 66 [1,1,1,0,0,0,1,0,1,0] => 24 [1,1,1,0,0,0,1,1,0,0] => 43 [1,1,1,0,0,1,0,0,1,0] => 27 [1,1,1,0,0,1,0,1,0,0] => 37 [1,1,1,0,0,1,1,0,0,0] => 57 [1,1,1,0,1,0,0,0,1,0] => 33 [1,1,1,0,1,0,0,1,0,0] => 43 [1,1,1,0,1,0,1,0,0,0] => 52 [1,1,1,0,1,1,0,0,0,0] => 76 [1,1,1,1,0,0,0,0,1,0] => 56 [1,1,1,1,0,0,0,1,0,0] => 66 [1,1,1,1,0,0,1,0,0,0] => 76 [1,1,1,1,0,1,0,0,0,0] => 90 [1,1,1,1,1,0,0,0,0,0] => 132 ----------------------------------------------------------------------------- Created: Aug 25, 2017 at 10:52 by Rene Marczinzik ----------------------------------------------------------------------------- Last Updated: Aug 25, 2020 at 15:00 by Martin Rubey