***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew <info@findstat.org> * * * * 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: St001278 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra. See theorem 5.8. in the reference for a motivation. ----------------------------------------------------------------------------- References: [1] Iyama, O., Solberg, Øyvind Auslander-Gorenstein algebras and precluster tilting. [[zbMATH:06833443]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1,0] => 0 [1,0,1,0] => 1 [1,1,0,0] => 0 [1,0,1,0,1,0] => 2 [1,0,1,1,0,0] => 1 [1,1,0,0,1,0] => 1 [1,1,0,1,0,0] => 2 [1,1,1,0,0,0] => 0 [1,0,1,0,1,0,1,0] => 3 [1,0,1,0,1,1,0,0] => 2 [1,0,1,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,0] => 3 [1,0,1,1,1,0,0,0] => 1 [1,1,0,0,1,0,1,0] => 2 [1,1,0,0,1,1,0,0] => 1 [1,1,0,1,0,0,1,0] => 3 [1,1,0,1,0,1,0,0] => 4 [1,1,0,1,1,0,0,0] => 2 [1,1,1,0,0,0,1,0] => 1 [1,1,1,0,0,1,0,0] => 2 [1,1,1,0,1,0,0,0] => 3 [1,1,1,1,0,0,0,0] => 0 [1,0,1,0,1,0,1,0,1,0] => 4 [1,0,1,0,1,0,1,1,0,0] => 3 [1,0,1,0,1,1,0,0,1,0] => 3 [1,0,1,0,1,1,0,1,0,0] => 4 [1,0,1,0,1,1,1,0,0,0] => 2 [1,0,1,1,0,0,1,0,1,0] => 3 [1,0,1,1,0,0,1,1,0,0] => 2 [1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,1,0,0] => 5 [1,0,1,1,0,1,1,0,0,0] => 3 [1,0,1,1,1,0,0,0,1,0] => 2 [1,0,1,1,1,0,0,1,0,0] => 3 [1,0,1,1,1,0,1,0,0,0] => 4 [1,0,1,1,1,1,0,0,0,0] => 1 [1,1,0,0,1,0,1,0,1,0] => 3 [1,1,0,0,1,0,1,1,0,0] => 2 [1,1,0,0,1,1,0,0,1,0] => 2 [1,1,0,0,1,1,0,1,0,0] => 3 [1,1,0,0,1,1,1,0,0,0] => 1 [1,1,0,1,0,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,1,0,0,1,0] => 5 [1,1,0,1,0,1,0,1,0,0] => 6 [1,1,0,1,0,1,1,0,0,0] => 4 [1,1,0,1,1,0,0,0,1,0] => 3 [1,1,0,1,1,0,0,1,0,0] => 4 [1,1,0,1,1,0,1,0,0,0] => 5 [1,1,0,1,1,1,0,0,0,0] => 2 [1,1,1,0,0,0,1,0,1,0] => 2 [1,1,1,0,0,0,1,1,0,0] => 1 [1,1,1,0,0,1,0,0,1,0] => 3 [1,1,1,0,0,1,0,1,0,0] => 4 [1,1,1,0,0,1,1,0,0,0] => 2 [1,1,1,0,1,0,0,0,1,0] => 4 [1,1,1,0,1,0,0,1,0,0] => 5 [1,1,1,0,1,0,1,0,0,0] => 6 [1,1,1,0,1,1,0,0,0,0] => 3 [1,1,1,1,0,0,0,0,1,0] => 1 [1,1,1,1,0,0,0,1,0,0] => 2 [1,1,1,1,0,0,1,0,0,0] => 3 [1,1,1,1,0,1,0,0,0,0] => 4 [1,1,1,1,1,0,0,0,0,0] => 0 [1,0,1,0,1,0,1,0,1,0,1,0] => 5 [1,0,1,0,1,0,1,0,1,1,0,0] => 4 [1,0,1,0,1,0,1,1,0,0,1,0] => 4 [1,0,1,0,1,0,1,1,0,1,0,0] => 5 [1,0,1,0,1,0,1,1,1,0,0,0] => 3 [1,0,1,0,1,1,0,0,1,0,1,0] => 4 [1,0,1,0,1,1,0,0,1,1,0,0] => 3 [1,0,1,0,1,1,0,1,0,0,1,0] => 5 [1,0,1,0,1,1,0,1,0,1,0,0] => 6 [1,0,1,0,1,1,0,1,1,0,0,0] => 4 [1,0,1,0,1,1,1,0,0,0,1,0] => 3 [1,0,1,0,1,1,1,0,0,1,0,0] => 4 [1,0,1,0,1,1,1,0,1,0,0,0] => 5 [1,0,1,0,1,1,1,1,0,0,0,0] => 2 [1,0,1,1,0,0,1,0,1,0,1,0] => 4 [1,0,1,1,0,0,1,0,1,1,0,0] => 3 [1,0,1,1,0,0,1,1,0,0,1,0] => 3 [1,0,1,1,0,0,1,1,0,1,0,0] => 4 [1,0,1,1,0,0,1,1,1,0,0,0] => 2 [1,0,1,1,0,1,0,0,1,0,1,0] => 5 [1,0,1,1,0,1,0,0,1,1,0,0] => 4 [1,0,1,1,0,1,0,1,0,0,1,0] => 6 [1,0,1,1,0,1,0,1,0,1,0,0] => 7 [1,0,1,1,0,1,0,1,1,0,0,0] => 5 [1,0,1,1,0,1,1,0,0,0,1,0] => 4 [1,0,1,1,0,1,1,0,0,1,0,0] => 5 [1,0,1,1,0,1,1,0,1,0,0,0] => 6 [1,0,1,1,0,1,1,1,0,0,0,0] => 3 [1,0,1,1,1,0,0,0,1,0,1,0] => 3 [1,0,1,1,1,0,0,0,1,1,0,0] => 2 [1,0,1,1,1,0,0,1,0,0,1,0] => 4 [1,0,1,1,1,0,0,1,0,1,0,0] => 5 [1,0,1,1,1,0,0,1,1,0,0,0] => 3 [1,0,1,1,1,0,1,0,0,0,1,0] => 5 [1,0,1,1,1,0,1,0,0,1,0,0] => 6 [1,0,1,1,1,0,1,0,1,0,0,0] => 7 [1,0,1,1,1,0,1,1,0,0,0,0] => 4 [1,0,1,1,1,1,0,0,0,0,1,0] => 2 [1,0,1,1,1,1,0,0,0,1,0,0] => 3 [1,0,1,1,1,1,0,0,1,0,0,0] => 4 [1,0,1,1,1,1,0,1,0,0,0,0] => 5 [1,0,1,1,1,1,1,0,0,0,0,0] => 1 [1,1,0,0,1,0,1,0,1,0,1,0] => 4 [1,1,0,0,1,0,1,0,1,1,0,0] => 3 [1,1,0,0,1,0,1,1,0,0,1,0] => 3 [1,1,0,0,1,0,1,1,0,1,0,0] => 4 [1,1,0,0,1,0,1,1,1,0,0,0] => 2 [1,1,0,0,1,1,0,0,1,0,1,0] => 3 [1,1,0,0,1,1,0,0,1,1,0,0] => 2 [1,1,0,0,1,1,0,1,0,0,1,0] => 4 [1,1,0,0,1,1,0,1,0,1,0,0] => 5 [1,1,0,0,1,1,0,1,1,0,0,0] => 3 [1,1,0,0,1,1,1,0,0,0,1,0] => 2 [1,1,0,0,1,1,1,0,0,1,0,0] => 3 [1,1,0,0,1,1,1,0,1,0,0,0] => 4 [1,1,0,0,1,1,1,1,0,0,0,0] => 1 [1,1,0,1,0,0,1,0,1,0,1,0] => 5 [1,1,0,1,0,0,1,0,1,1,0,0] => 4 [1,1,0,1,0,0,1,1,0,0,1,0] => 4 [1,1,0,1,0,0,1,1,0,1,0,0] => 5 [1,1,0,1,0,0,1,1,1,0,0,0] => 3 [1,1,0,1,0,1,0,0,1,0,1,0] => 6 [1,1,0,1,0,1,0,0,1,1,0,0] => 5 [1,1,0,1,0,1,0,1,0,0,1,0] => 7 [1,1,0,1,0,1,0,1,0,1,0,0] => 8 [1,1,0,1,0,1,0,1,1,0,0,0] => 6 [1,1,0,1,0,1,1,0,0,0,1,0] => 5 [1,1,0,1,0,1,1,0,0,1,0,0] => 6 [1,1,0,1,0,1,1,0,1,0,0,0] => 7 [1,1,0,1,0,1,1,1,0,0,0,0] => 4 [1,1,0,1,1,0,0,0,1,0,1,0] => 4 [1,1,0,1,1,0,0,0,1,1,0,0] => 3 [1,1,0,1,1,0,0,1,0,0,1,0] => 5 [1,1,0,1,1,0,0,1,0,1,0,0] => 6 [1,1,0,1,1,0,0,1,1,0,0,0] => 4 [1,1,0,1,1,0,1,0,0,0,1,0] => 6 [1,1,0,1,1,0,1,0,0,1,0,0] => 7 [1,1,0,1,1,0,1,0,1,0,0,0] => 8 [1,1,0,1,1,0,1,1,0,0,0,0] => 5 [1,1,0,1,1,1,0,0,0,0,1,0] => 3 [1,1,0,1,1,1,0,0,0,1,0,0] => 4 [1,1,0,1,1,1,0,0,1,0,0,0] => 5 [1,1,0,1,1,1,0,1,0,0,0,0] => 6 [1,1,0,1,1,1,1,0,0,0,0,0] => 2 [1,1,1,0,0,0,1,0,1,0,1,0] => 3 [1,1,1,0,0,0,1,0,1,1,0,0] => 2 [1,1,1,0,0,0,1,1,0,0,1,0] => 2 [1,1,1,0,0,0,1,1,0,1,0,0] => 3 [1,1,1,0,0,0,1,1,1,0,0,0] => 1 [1,1,1,0,0,1,0,0,1,0,1,0] => 4 [1,1,1,0,0,1,0,0,1,1,0,0] => 3 [1,1,1,0,0,1,0,1,0,0,1,0] => 5 [1,1,1,0,0,1,0,1,0,1,0,0] => 6 [1,1,1,0,0,1,0,1,1,0,0,0] => 4 [1,1,1,0,0,1,1,0,0,0,1,0] => 3 [1,1,1,0,0,1,1,0,0,1,0,0] => 4 [1,1,1,0,0,1,1,0,1,0,0,0] => 5 [1,1,1,0,0,1,1,1,0,0,0,0] => 2 [1,1,1,0,1,0,0,0,1,0,1,0] => 5 [1,1,1,0,1,0,0,0,1,1,0,0] => 4 [1,1,1,0,1,0,0,1,0,0,1,0] => 6 [1,1,1,0,1,0,0,1,0,1,0,0] => 7 [1,1,1,0,1,0,0,1,1,0,0,0] => 5 [1,1,1,0,1,0,1,0,0,0,1,0] => 7 [1,1,1,0,1,0,1,0,0,1,0,0] => 8 [1,1,1,0,1,0,1,0,1,0,0,0] => 9 [1,1,1,0,1,0,1,1,0,0,0,0] => 6 [1,1,1,0,1,1,0,0,0,0,1,0] => 4 [1,1,1,0,1,1,0,0,0,1,0,0] => 5 [1,1,1,0,1,1,0,0,1,0,0,0] => 6 [1,1,1,0,1,1,0,1,0,0,0,0] => 7 [1,1,1,0,1,1,1,0,0,0,0,0] => 3 [1,1,1,1,0,0,0,0,1,0,1,0] => 2 [1,1,1,1,0,0,0,0,1,1,0,0] => 1 [1,1,1,1,0,0,0,1,0,0,1,0] => 3 [1,1,1,1,0,0,0,1,0,1,0,0] => 4 [1,1,1,1,0,0,0,1,1,0,0,0] => 2 [1,1,1,1,0,0,1,0,0,0,1,0] => 4 [1,1,1,1,0,0,1,0,0,1,0,0] => 5 [1,1,1,1,0,0,1,0,1,0,0,0] => 6 [1,1,1,1,0,0,1,1,0,0,0,0] => 3 [1,1,1,1,0,1,0,0,0,0,1,0] => 5 [1,1,1,1,0,1,0,0,0,1,0,0] => 6 [1,1,1,1,0,1,0,0,1,0,0,0] => 7 [1,1,1,1,0,1,0,1,0,0,0,0] => 8 [1,1,1,1,0,1,1,0,0,0,0,0] => 4 [1,1,1,1,1,0,0,0,0,0,1,0] => 1 [1,1,1,1,1,0,0,0,0,1,0,0] => 2 [1,1,1,1,1,0,0,0,1,0,0,0] => 3 [1,1,1,1,1,0,0,1,0,0,0,0] => 4 [1,1,1,1,1,0,1,0,0,0,0,0] => 5 [1,1,1,1,1,1,0,0,0,0,0,0] => 0 ----------------------------------------------------------------------------- Created: Oct 20, 2018 at 20:09 by Rene Marczinzik ----------------------------------------------------------------------------- Last Updated: Nov 26, 2018 at 21:48 by Rene Marczinzik