***************************************************************************** * 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: St001873 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). The statistic gives half of the rank of the matrix C^t-C. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1,0] => 0 [1,0,1,0] => 0 [1,1,0,0] => 1 [1,0,1,0,1,0] => 0 [1,0,1,1,0,0] => 1 [1,1,0,0,1,0] => 1 [1,1,0,1,0,0] => 1 [1,1,1,0,0,0] => 1 [1,0,1,0,1,0,1,0] => 0 [1,0,1,0,1,1,0,0] => 1 [1,0,1,1,0,0,1,0] => 1 [1,0,1,1,0,1,0,0] => 1 [1,0,1,1,1,0,0,0] => 1 [1,1,0,0,1,0,1,0] => 1 [1,1,0,0,1,1,0,0] => 2 [1,1,0,1,0,0,1,0] => 1 [1,1,0,1,0,1,0,0] => 2 [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] => 1 [1,1,1,1,0,0,0,0] => 2 [1,0,1,0,1,0,1,0,1,0] => 0 [1,0,1,0,1,0,1,1,0,0] => 1 [1,0,1,0,1,1,0,0,1,0] => 1 [1,0,1,0,1,1,0,1,0,0] => 1 [1,0,1,0,1,1,1,0,0,0] => 1 [1,0,1,1,0,0,1,0,1,0] => 1 [1,0,1,1,0,0,1,1,0,0] => 2 [1,0,1,1,0,1,0,0,1,0] => 1 [1,0,1,1,0,1,0,1,0,0] => 2 [1,0,1,1,0,1,1,0,0,0] => 2 [1,0,1,1,1,0,0,0,1,0] => 1 [1,0,1,1,1,0,0,1,0,0] => 2 [1,0,1,1,1,0,1,0,0,0] => 1 [1,0,1,1,1,1,0,0,0,0] => 2 [1,1,0,0,1,0,1,0,1,0] => 1 [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] => 2 [1,1,0,0,1,1,1,0,0,0] => 2 [1,1,0,1,0,0,1,0,1,0] => 1 [1,1,0,1,0,0,1,1,0,0] => 2 [1,1,0,1,0,1,0,0,1,0] => 2 [1,1,0,1,0,1,0,1,0,0] => 2 [1,1,0,1,0,1,1,0,0,0] => 2 [1,1,0,1,1,0,0,0,1,0] => 2 [1,1,0,1,1,0,0,1,0,0] => 2 [1,1,0,1,1,0,1,0,0,0] => 2 [1,1,0,1,1,1,0,0,0,0] => 2 [1,1,1,0,0,0,1,0,1,0] => 1 [1,1,1,0,0,0,1,1,0,0] => 2 [1,1,1,0,0,1,0,0,1,0] => 2 [1,1,1,0,0,1,0,1,0,0] => 2 [1,1,1,0,0,1,1,0,0,0] => 2 [1,1,1,0,1,0,0,0,1,0] => 1 [1,1,1,0,1,0,0,1,0,0] => 2 [1,1,1,0,1,0,1,0,0,0] => 2 [1,1,1,0,1,1,0,0,0,0] => 2 [1,1,1,1,0,0,0,0,1,0] => 2 [1,1,1,1,0,0,0,1,0,0] => 2 [1,1,1,1,0,0,1,0,0,0] => 2 [1,1,1,1,0,1,0,0,0,0] => 2 [1,1,1,1,1,0,0,0,0,0] => 2 [1,0,1,0,1,0,1,0,1,0,1,0] => 0 [1,0,1,0,1,0,1,0,1,1,0,0] => 1 [1,0,1,0,1,0,1,1,0,0,1,0] => 1 [1,0,1,0,1,0,1,1,0,1,0,0] => 1 [1,0,1,0,1,0,1,1,1,0,0,0] => 1 [1,0,1,0,1,1,0,0,1,0,1,0] => 1 [1,0,1,0,1,1,0,0,1,1,0,0] => 2 [1,0,1,0,1,1,0,1,0,0,1,0] => 1 [1,0,1,0,1,1,0,1,0,1,0,0] => 2 [1,0,1,0,1,1,0,1,1,0,0,0] => 2 [1,0,1,0,1,1,1,0,0,0,1,0] => 1 [1,0,1,0,1,1,1,0,0,1,0,0] => 2 [1,0,1,0,1,1,1,0,1,0,0,0] => 1 [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] => 1 [1,0,1,1,0,0,1,0,1,1,0,0] => 2 [1,0,1,1,0,0,1,1,0,0,1,0] => 2 [1,0,1,1,0,0,1,1,0,1,0,0] => 2 [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] => 1 [1,0,1,1,0,1,0,0,1,1,0,0] => 2 [1,0,1,1,0,1,0,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,1,0,1,0,0] => 2 [1,0,1,1,0,1,0,1,1,0,0,0] => 2 [1,0,1,1,0,1,1,0,0,0,1,0] => 2 [1,0,1,1,0,1,1,0,0,1,0,0] => 2 [1,0,1,1,0,1,1,0,1,0,0,0] => 2 [1,0,1,1,0,1,1,1,0,0,0,0] => 2 [1,0,1,1,1,0,0,0,1,0,1,0] => 1 [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] => 2 [1,0,1,1,1,0,0,1,0,1,0,0] => 2 [1,0,1,1,1,0,0,1,1,0,0,0] => 2 [1,0,1,1,1,0,1,0,0,0,1,0] => 1 [1,0,1,1,1,0,1,0,0,1,0,0] => 2 [1,0,1,1,1,0,1,0,1,0,0,0] => 2 [1,0,1,1,1,0,1,1,0,0,0,0] => 2 [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] => 2 [1,0,1,1,1,1,0,0,1,0,0,0] => 2 [1,0,1,1,1,1,0,1,0,0,0,0] => 2 [1,0,1,1,1,1,1,0,0,0,0,0] => 2 [1,1,0,0,1,0,1,0,1,0,1,0] => 1 [1,1,0,0,1,0,1,0,1,1,0,0] => 2 [1,1,0,0,1,0,1,1,0,0,1,0] => 2 [1,1,0,0,1,0,1,1,0,1,0,0] => 2 [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] => 2 [1,1,0,0,1,1,0,0,1,1,0,0] => 3 [1,1,0,0,1,1,0,1,0,0,1,0] => 2 [1,1,0,0,1,1,0,1,0,1,0,0] => 3 [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] => 2 [1,1,0,0,1,1,1,1,0,0,0,0] => 3 [1,1,0,1,0,0,1,0,1,0,1,0] => 1 [1,1,0,1,0,0,1,0,1,1,0,0] => 2 [1,1,0,1,0,0,1,1,0,0,1,0] => 2 [1,1,0,1,0,0,1,1,0,1,0,0] => 2 [1,1,0,1,0,0,1,1,1,0,0,0] => 2 [1,1,0,1,0,1,0,0,1,0,1,0] => 2 [1,1,0,1,0,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,1,0,1,0,0,1,0] => 2 [1,1,0,1,0,1,0,1,0,1,0,0] => 3 [1,1,0,1,0,1,0,1,1,0,0,0] => 3 [1,1,0,1,0,1,1,0,0,0,1,0] => 2 [1,1,0,1,0,1,1,0,0,1,0,0] => 3 [1,1,0,1,0,1,1,0,1,0,0,0] => 2 [1,1,0,1,0,1,1,1,0,0,0,0] => 3 [1,1,0,1,1,0,0,0,1,0,1,0] => 2 [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] => 2 [1,1,0,1,1,0,0,1,0,1,0,0] => 3 [1,1,0,1,1,0,0,1,1,0,0,0] => 3 [1,1,0,1,1,0,1,0,0,0,1,0] => 2 [1,1,0,1,1,0,1,0,0,1,0,0] => 3 [1,1,0,1,1,0,1,0,1,0,0,0] => 2 [1,1,0,1,1,0,1,1,0,0,0,0] => 3 [1,1,0,1,1,1,0,0,0,0,1,0] => 2 [1,1,0,1,1,1,0,0,0,1,0,0] => 3 [1,1,0,1,1,1,0,0,1,0,0,0] => 2 [1,1,0,1,1,1,0,1,0,0,0,0] => 3 [1,1,0,1,1,1,1,0,0,0,0,0] => 3 [1,1,1,0,0,0,1,0,1,0,1,0] => 1 [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] => 2 [1,1,1,0,0,0,1,1,1,0,0,0] => 2 [1,1,1,0,0,1,0,0,1,0,1,0] => 2 [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] => 2 [1,1,1,0,0,1,0,1,0,1,0,0] => 3 [1,1,1,0,0,1,0,1,1,0,0,0] => 3 [1,1,1,0,0,1,1,0,0,0,1,0] => 2 [1,1,1,0,0,1,1,0,0,1,0,0] => 3 [1,1,1,0,0,1,1,0,1,0,0,0] => 2 [1,1,1,0,0,1,1,1,0,0,0,0] => 3 [1,1,1,0,1,0,0,0,1,0,1,0] => 1 [1,1,1,0,1,0,0,0,1,1,0,0] => 2 [1,1,1,0,1,0,0,1,0,0,1,0] => 2 [1,1,1,0,1,0,0,1,0,1,0,0] => 2 [1,1,1,0,1,0,0,1,1,0,0,0] => 2 [1,1,1,0,1,0,1,0,0,0,1,0] => 2 [1,1,1,0,1,0,1,0,0,1,0,0] => 2 [1,1,1,0,1,0,1,0,1,0,0,0] => 3 [1,1,1,0,1,0,1,1,0,0,0,0] => 2 [1,1,1,0,1,1,0,0,0,0,1,0] => 2 [1,1,1,0,1,1,0,0,0,1,0,0] => 2 [1,1,1,0,1,1,0,0,1,0,0,0] => 2 [1,1,1,0,1,1,0,1,0,0,0,0] => 3 [1,1,1,0,1,1,1,0,0,0,0,0] => 2 [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] => 3 [1,1,1,1,0,0,0,1,0,0,1,0] => 2 [1,1,1,1,0,0,0,1,0,1,0,0] => 3 [1,1,1,1,0,0,0,1,1,0,0,0] => 3 [1,1,1,1,0,0,1,0,0,0,1,0] => 2 [1,1,1,1,0,0,1,0,0,1,0,0] => 3 [1,1,1,1,0,0,1,0,1,0,0,0] => 2 [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] => 2 [1,1,1,1,0,1,0,0,0,1,0,0] => 3 [1,1,1,1,0,1,0,0,1,0,0,0] => 3 [1,1,1,1,0,1,0,1,0,0,0,0] => 3 [1,1,1,1,0,1,1,0,0,0,0,0] => 3 [1,1,1,1,1,0,0,0,0,0,1,0] => 2 [1,1,1,1,1,0,0,0,0,1,0,0] => 3 [1,1,1,1,1,0,0,0,1,0,0,0] => 2 [1,1,1,1,1,0,0,1,0,0,0,0] => 3 [1,1,1,1,1,0,1,0,0,0,0,0] => 2 [1,1,1,1,1,1,0,0,0,0,0,0] => 3 ----------------------------------------------------------------------------- Created: Jan 05, 2020 at 16:56 by Rene Marczinzik ----------------------------------------------------------------------------- Last Updated: Jan 05, 2020 at 16:56 by Rene Marczinzik