***************************************************************************** * 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: St001491 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: 1 => 1 01 => 1 10 => 1 11 => 2 001 => 1 010 => 1 011 => 1 100 => 1 101 => 2 110 => 1 111 => 3 0001 => 1 0010 => 1 0011 => 1 0100 => 1 0101 => 0 0110 => 2 0111 => 2 1000 => 1 1001 => 2 1010 => 0 1011 => 2 1100 => 1 1101 => 2 1110 => 2 1111 => 4 ----------------------------------------------------------------------------- Created: Nov 09, 2019 at 14:36 by Rene Marczinzik ----------------------------------------------------------------------------- Last Updated: Nov 09, 2019 at 15:11 by Rene Marczinzik