***************************************************************************** * 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: St000135 ----------------------------------------------------------------------------- Collection: Parking functions ----------------------------------------------------------------------------- Description: The number of lucky cars of the parking function. A lucky car is a car that was able to park in its prefered spot. The generating function, $$ q\prod_{i=1}^{n-1} (i + (n-i+1)q) $$ was established in [1]. ----------------------------------------------------------------------------- References: [1] Gessel, I. M., Seo, S. A refinement of Cayley's formula for trees [[MathSciNet:2224940]] ----------------------------------------------------------------------------- Code: def statistic(pf): return len(pf.lucky_cars()) def generating_function(n): R. = ZZ[] if n: return q * prod(i + q*(n-i+1) for i in range(1, n)) return R.one() ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,1] => 1 [1,2] => 2 [2,1] => 2 [1,1,1] => 1 [1,1,2] => 1 [1,2,1] => 2 [2,1,1] => 2 [1,1,3] => 2 [1,3,1] => 2 [3,1,1] => 2 [1,2,2] => 2 [2,1,2] => 2 [2,2,1] => 2 [1,2,3] => 3 [1,3,2] => 3 [2,1,3] => 3 [2,3,1] => 3 [3,1,2] => 3 [3,2,1] => 3 [1,1,1,1] => 1 [1,1,1,2] => 1 [1,1,2,1] => 1 [1,2,1,1] => 2 [2,1,1,1] => 2 [1,1,1,3] => 1 [1,1,3,1] => 2 [1,3,1,1] => 2 [3,1,1,1] => 2 [1,1,1,4] => 2 [1,1,4,1] => 2 [1,4,1,1] => 2 [4,1,1,1] => 2 [1,1,2,2] => 1 [1,2,1,2] => 2 [1,2,2,1] => 2 [2,1,1,2] => 2 [2,1,2,1] => 2 [2,2,1,1] => 2 [1,1,2,3] => 1 [1,1,3,2] => 2 [1,2,1,3] => 2 [1,2,3,1] => 3 [1,3,1,2] => 2 [1,3,2,1] => 3 [2,1,1,3] => 2 [2,1,3,1] => 3 [2,3,1,1] => 3 [3,1,1,2] => 2 [3,1,2,1] => 3 [3,2,1,1] => 3 [1,1,2,4] => 2 [1,1,4,2] => 2 [1,2,1,4] => 3 [1,2,4,1] => 3 [1,4,1,2] => 2 [1,4,2,1] => 3 [2,1,1,4] => 3 [2,1,4,1] => 3 [2,4,1,1] => 3 [4,1,1,2] => 2 [4,1,2,1] => 3 [4,2,1,1] => 3 [1,1,3,3] => 2 [1,3,1,3] => 2 [1,3,3,1] => 2 [3,1,1,3] => 2 [3,1,3,1] => 2 [3,3,1,1] => 2 [1,1,3,4] => 3 [1,1,4,3] => 3 [1,3,1,4] => 3 [1,3,4,1] => 3 [1,4,1,3] => 3 [1,4,3,1] => 3 [3,1,1,4] => 3 [3,1,4,1] => 3 [3,4,1,1] => 3 [4,1,1,3] => 3 [4,1,3,1] => 3 [4,3,1,1] => 3 [1,2,2,2] => 2 [2,1,2,2] => 2 [2,2,1,2] => 2 [2,2,2,1] => 2 [1,2,2,3] => 2 [1,2,3,2] => 3 [1,3,2,2] => 3 [2,1,2,3] => 2 [2,1,3,2] => 3 [2,2,1,3] => 2 [2,2,3,1] => 2 [2,3,1,2] => 3 [2,3,2,1] => 3 [3,1,2,2] => 3 [3,2,1,2] => 3 [3,2,2,1] => 3 [1,2,2,4] => 3 [1,2,4,2] => 3 [1,4,2,2] => 3 [2,1,2,4] => 3 [2,1,4,2] => 3 [2,2,1,4] => 3 [2,2,4,1] => 3 [2,4,1,2] => 3 [2,4,2,1] => 3 [4,1,2,2] => 3 [4,2,1,2] => 3 [4,2,2,1] => 3 [1,2,3,3] => 3 [1,3,2,3] => 3 [1,3,3,2] => 3 [2,1,3,3] => 3 [2,3,1,3] => 3 [2,3,3,1] => 3 [3,1,2,3] => 3 [3,1,3,2] => 3 [3,2,1,3] => 3 [3,2,3,1] => 3 [3,3,1,2] => 3 [3,3,2,1] => 3 [1,2,3,4] => 4 [1,2,4,3] => 4 [1,3,2,4] => 4 [1,3,4,2] => 4 [1,4,2,3] => 4 [1,4,3,2] => 4 [2,1,3,4] => 4 [2,1,4,3] => 4 [2,3,1,4] => 4 [2,3,4,1] => 4 [2,4,1,3] => 4 [2,4,3,1] => 4 [3,1,2,4] => 4 [3,1,4,2] => 4 [3,2,1,4] => 4 [3,2,4,1] => 4 [3,4,1,2] => 4 [3,4,2,1] => 4 [4,1,2,3] => 4 [4,1,3,2] => 4 [4,2,1,3] => 4 [4,2,3,1] => 4 [4,3,1,2] => 4 [4,3,2,1] => 4 ----------------------------------------------------------------------------- Created: Jun 20, 2013 at 11:13 by Viviane Pons ----------------------------------------------------------------------------- Last Updated: Oct 11, 2024 at 10:55 by Martin Rubey