***************************************************************************** * 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: St001890 ----------------------------------------------------------------------------- Collection: Posets ----------------------------------------------------------------------------- Description: The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Incidence_algebra#Special_elements]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ([],2) => 1 ([(0,1)],2) => 1 ([],3) => 1 ([(1,2)],3) => 1 ([(0,1),(0,2)],3) => 1 ([(0,2),(2,1)],3) => 1 ([(0,2),(1,2)],3) => 1 ([],4) => 1 ([(2,3)],4) => 1 ([(1,2),(1,3)],4) => 1 ([(0,1),(0,2),(0,3)],4) => 1 ([(0,2),(0,3),(3,1)],4) => 1 ([(0,1),(0,2),(1,3),(2,3)],4) => 1 ([(1,2),(2,3)],4) => 1 ([(0,3),(3,1),(3,2)],4) => 1 ([(1,3),(2,3)],4) => 1 ([(0,3),(1,3),(3,2)],4) => 1 ([(0,3),(1,3),(2,3)],4) => 1 ([(0,3),(1,2)],4) => 1 ([(0,3),(1,2),(1,3)],4) => 1 ([(0,2),(0,3),(1,2),(1,3)],4) => 1 ([(0,3),(2,1),(3,2)],4) => 1 ([(0,3),(1,2),(2,3)],4) => 1 ([],5) => 1 ([(3,4)],5) => 1 ([(2,3),(2,4)],5) => 1 ([(1,2),(1,3),(1,4)],5) => 1 ([(0,1),(0,2),(0,3),(0,4)],5) => 1 ([(0,2),(0,3),(0,4),(4,1)],5) => 1 ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1 ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2 ([(1,3),(1,4),(4,2)],5) => 1 ([(0,3),(0,4),(4,1),(4,2)],5) => 1 ([(1,2),(1,3),(2,4),(3,4)],5) => 1 ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1 ([(0,3),(0,4),(3,2),(4,1)],5) => 1 ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1 ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 1 ([(2,3),(3,4)],5) => 1 ([(1,4),(4,2),(4,3)],5) => 1 ([(0,4),(4,1),(4,2),(4,3)],5) => 1 ([(2,4),(3,4)],5) => 1 ([(1,4),(2,4),(4,3)],5) => 1 ([(0,4),(1,4),(4,2),(4,3)],5) => 1 ([(1,4),(2,4),(3,4)],5) => 1 ([(0,4),(1,4),(2,4),(4,3)],5) => 1 ([(0,4),(1,4),(2,4),(3,4)],5) => 1 ([(0,4),(1,4),(2,3)],5) => 1 ([(0,4),(1,3),(2,3),(2,4)],5) => 1 ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1 ([(0,4),(1,4),(2,3),(4,2)],5) => 1 ([(0,4),(1,3),(2,3),(3,4)],5) => 1 ([(0,4),(1,4),(2,3),(2,4)],5) => 1 ([(0,4),(1,4),(2,3),(3,4)],5) => 1 ([(1,4),(2,3)],5) => 1 ([(1,4),(2,3),(2,4)],5) => 1 ([(0,4),(1,2),(1,4),(2,3)],5) => 1 ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1 ([(1,3),(1,4),(2,3),(2,4)],5) => 1 ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 1 ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 1 ([(0,4),(1,2),(1,4),(4,3)],5) => 1 ([(0,4),(1,2),(1,3)],5) => 1 ([(0,4),(1,2),(1,3),(1,4)],5) => 1 ([(0,2),(0,4),(3,1),(4,3)],5) => 1 ([(0,4),(1,2),(1,3),(3,4)],5) => 1 ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1 ([(0,3),(0,4),(1,2),(1,4)],5) => 1 ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1 ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 1 ([(0,3),(1,2),(1,4),(3,4)],5) => 1 ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 1 ([(1,4),(3,2),(4,3)],5) => 1 ([(0,3),(3,4),(4,1),(4,2)],5) => 1 ([(1,4),(2,3),(3,4)],5) => 1 ([(0,4),(1,2),(2,4),(4,3)],5) => 1 ([(0,3),(1,4),(4,2)],5) => 1 ([(0,4),(3,2),(4,1),(4,3)],5) => 1 ([(0,4),(1,2),(2,3),(2,4)],5) => 1 ([(0,4),(2,3),(3,1),(4,2)],5) => 1 ([(0,3),(1,2),(2,4),(3,4)],5) => 1 ([(0,4),(1,2),(2,3),(3,4)],5) => 1 ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1 ----------------------------------------------------------------------------- Created: Mar 13, 2023 at 22:17 by Harry Richman ----------------------------------------------------------------------------- Last Updated: Mar 13, 2023 at 22:17 by Harry Richman