***************************************************************************** * 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: St000551 ----------------------------------------------------------------------------- Collection: Lattices ----------------------------------------------------------------------------- Description: The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$. ----------------------------------------------------------------------------- References: [1] Liu, S.-C., Sagan, B. Left-modular elements [[arXiv:math/0001055]] ----------------------------------------------------------------------------- Code: def is_left_modular(L, x): for z in L: for y in L.principal_order_ideal(z): if L.meet(L.join(x,y),z) != L.join(L.meet(x,z),y): return False return True def statistic(L): return len([x for x in L if is_left_modular(L, x)]) ----------------------------------------------------------------------------- Statistic values: ([],1) => 1 ([(0,1)],2) => 2 ([(0,2),(2,1)],3) => 3 ([(0,1),(0,2),(1,3),(2,3)],4) => 4 ([(0,3),(2,1),(3,2)],4) => 4 ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5 ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4 ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5 ([(0,4),(2,3),(3,1),(4,2)],5) => 5 ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5 ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 6 ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 4 ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 5 ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 6 ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5 ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6 ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 5 ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6 ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 5 ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6 ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 6 ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 2 ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6 ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6 ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 7 ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 4 ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 5 ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 6 ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 7 ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 5 ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 6 ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 7 ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 5 ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 6 ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 6 ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 7 ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 3 ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 7 ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 5 ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 6 ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 7 ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 5 ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 6 ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 7 ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 6 ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 7 ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 7 ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 2 ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 5 ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 7 ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 6 ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 7 ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 5 ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 2 ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 6 ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 5 ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 7 ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6 ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2 ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 7 ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 6 ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 5 ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6 ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 6 ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 5 ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 5 ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 6 ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7 ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 5 ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6 ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 2 ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6 ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6 ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7 ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 3 ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 6 ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7 ----------------------------------------------------------------------------- Created: Jul 19, 2016 at 23:49 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 21, 2016 at 10:49 by Martin Rubey