***************************************************************************** * 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: St000014 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The number of parking functions supported by a Dyck path. One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition. ----------------------------------------------------------------------------- References: [1] [[http://www.findstat.org/ParkingFunctions]] ----------------------------------------------------------------------------- Code: def statistic(x): return x.number_of_parking_functions() ----------------------------------------------------------------------------- Statistic values: [1,0] => 1 [1,0,1,0] => 2 [1,1,0,0] => 1 [1,0,1,0,1,0] => 6 [1,0,1,1,0,0] => 3 [1,1,0,0,1,0] => 3 [1,1,0,1,0,0] => 3 [1,1,1,0,0,0] => 1 [1,0,1,0,1,0,1,0] => 24 [1,0,1,0,1,1,0,0] => 12 [1,0,1,1,0,0,1,0] => 12 [1,0,1,1,0,1,0,0] => 12 [1,0,1,1,1,0,0,0] => 4 [1,1,0,0,1,0,1,0] => 12 [1,1,0,0,1,1,0,0] => 6 [1,1,0,1,0,0,1,0] => 12 [1,1,0,1,0,1,0,0] => 12 [1,1,0,1,1,0,0,0] => 6 [1,1,1,0,0,0,1,0] => 4 [1,1,1,0,0,1,0,0] => 4 [1,1,1,0,1,0,0,0] => 4 [1,1,1,1,0,0,0,0] => 1 [1,0,1,0,1,0,1,0,1,0] => 120 [1,0,1,0,1,0,1,1,0,0] => 60 [1,0,1,0,1,1,0,0,1,0] => 60 [1,0,1,0,1,1,0,1,0,0] => 60 [1,0,1,0,1,1,1,0,0,0] => 20 [1,0,1,1,0,0,1,0,1,0] => 60 [1,0,1,1,0,0,1,1,0,0] => 30 [1,0,1,1,0,1,0,0,1,0] => 60 [1,0,1,1,0,1,0,1,0,0] => 60 [1,0,1,1,0,1,1,0,0,0] => 30 [1,0,1,1,1,0,0,0,1,0] => 20 [1,0,1,1,1,0,0,1,0,0] => 20 [1,0,1,1,1,0,1,0,0,0] => 20 [1,0,1,1,1,1,0,0,0,0] => 5 [1,1,0,0,1,0,1,0,1,0] => 60 [1,1,0,0,1,0,1,1,0,0] => 30 [1,1,0,0,1,1,0,0,1,0] => 30 [1,1,0,0,1,1,0,1,0,0] => 30 [1,1,0,0,1,1,1,0,0,0] => 10 [1,1,0,1,0,0,1,0,1,0] => 60 [1,1,0,1,0,0,1,1,0,0] => 30 [1,1,0,1,0,1,0,0,1,0] => 60 [1,1,0,1,0,1,0,1,0,0] => 60 [1,1,0,1,0,1,1,0,0,0] => 30 [1,1,0,1,1,0,0,0,1,0] => 30 [1,1,0,1,1,0,0,1,0,0] => 30 [1,1,0,1,1,0,1,0,0,0] => 30 [1,1,0,1,1,1,0,0,0,0] => 10 [1,1,1,0,0,0,1,0,1,0] => 20 [1,1,1,0,0,0,1,1,0,0] => 10 [1,1,1,0,0,1,0,0,1,0] => 20 [1,1,1,0,0,1,0,1,0,0] => 20 [1,1,1,0,0,1,1,0,0,0] => 10 [1,1,1,0,1,0,0,0,1,0] => 20 [1,1,1,0,1,0,0,1,0,0] => 20 [1,1,1,0,1,0,1,0,0,0] => 20 [1,1,1,0,1,1,0,0,0,0] => 10 [1,1,1,1,0,0,0,0,1,0] => 5 [1,1,1,1,0,0,0,1,0,0] => 5 [1,1,1,1,0,0,1,0,0,0] => 5 [1,1,1,1,0,1,0,0,0,0] => 5 [1,1,1,1,1,0,0,0,0,0] => 1 [1,0,1,0,1,0,1,0,1,0,1,0] => 720 [1,0,1,0,1,0,1,0,1,1,0,0] => 360 [1,0,1,0,1,0,1,1,0,0,1,0] => 360 [1,0,1,0,1,0,1,1,0,1,0,0] => 360 [1,0,1,0,1,0,1,1,1,0,0,0] => 120 [1,0,1,0,1,1,0,0,1,0,1,0] => 360 [1,0,1,0,1,1,0,0,1,1,0,0] => 180 [1,0,1,0,1,1,0,1,0,0,1,0] => 360 [1,0,1,0,1,1,0,1,0,1,0,0] => 360 [1,0,1,0,1,1,0,1,1,0,0,0] => 180 [1,0,1,0,1,1,1,0,0,0,1,0] => 120 [1,0,1,0,1,1,1,0,0,1,0,0] => 120 [1,0,1,0,1,1,1,0,1,0,0,0] => 120 [1,0,1,0,1,1,1,1,0,0,0,0] => 30 [1,0,1,1,0,0,1,0,1,0,1,0] => 360 [1,0,1,1,0,0,1,0,1,1,0,0] => 180 [1,0,1,1,0,0,1,1,0,0,1,0] => 180 [1,0,1,1,0,0,1,1,0,1,0,0] => 180 [1,0,1,1,0,0,1,1,1,0,0,0] => 60 [1,0,1,1,0,1,0,0,1,0,1,0] => 360 [1,0,1,1,0,1,0,0,1,1,0,0] => 180 [1,0,1,1,0,1,0,1,0,0,1,0] => 360 [1,0,1,1,0,1,0,1,0,1,0,0] => 360 [1,0,1,1,0,1,0,1,1,0,0,0] => 180 [1,0,1,1,0,1,1,0,0,0,1,0] => 180 [1,0,1,1,0,1,1,0,0,1,0,0] => 180 [1,0,1,1,0,1,1,0,1,0,0,0] => 180 [1,0,1,1,0,1,1,1,0,0,0,0] => 60 [1,0,1,1,1,0,0,0,1,0,1,0] => 120 [1,0,1,1,1,0,0,0,1,1,0,0] => 60 [1,0,1,1,1,0,0,1,0,0,1,0] => 120 [1,0,1,1,1,0,0,1,0,1,0,0] => 120 [1,0,1,1,1,0,0,1,1,0,0,0] => 60 [1,0,1,1,1,0,1,0,0,0,1,0] => 120 [1,0,1,1,1,0,1,0,0,1,0,0] => 120 [1,0,1,1,1,0,1,0,1,0,0,0] => 120 [1,0,1,1,1,0,1,1,0,0,0,0] => 60 [1,0,1,1,1,1,0,0,0,0,1,0] => 30 [1,0,1,1,1,1,0,0,0,1,0,0] => 30 [1,0,1,1,1,1,0,0,1,0,0,0] => 30 [1,0,1,1,1,1,0,1,0,0,0,0] => 30 [1,0,1,1,1,1,1,0,0,0,0,0] => 6 [1,1,0,0,1,0,1,0,1,0,1,0] => 360 [1,1,0,0,1,0,1,0,1,1,0,0] => 180 [1,1,0,0,1,0,1,1,0,0,1,0] => 180 [1,1,0,0,1,0,1,1,0,1,0,0] => 180 [1,1,0,0,1,0,1,1,1,0,0,0] => 60 [1,1,0,0,1,1,0,0,1,0,1,0] => 180 [1,1,0,0,1,1,0,0,1,1,0,0] => 90 [1,1,0,0,1,1,0,1,0,0,1,0] => 180 [1,1,0,0,1,1,0,1,0,1,0,0] => 180 [1,1,0,0,1,1,0,1,1,0,0,0] => 90 [1,1,0,0,1,1,1,0,0,0,1,0] => 60 [1,1,0,0,1,1,1,0,0,1,0,0] => 60 [1,1,0,0,1,1,1,0,1,0,0,0] => 60 [1,1,0,0,1,1,1,1,0,0,0,0] => 15 [1,1,0,1,0,0,1,0,1,0,1,0] => 360 [1,1,0,1,0,0,1,0,1,1,0,0] => 180 [1,1,0,1,0,0,1,1,0,0,1,0] => 180 [1,1,0,1,0,0,1,1,0,1,0,0] => 180 [1,1,0,1,0,0,1,1,1,0,0,0] => 60 [1,1,0,1,0,1,0,0,1,0,1,0] => 360 [1,1,0,1,0,1,0,0,1,1,0,0] => 180 [1,1,0,1,0,1,0,1,0,0,1,0] => 360 [1,1,0,1,0,1,0,1,0,1,0,0] => 360 [1,1,0,1,0,1,0,1,1,0,0,0] => 180 [1,1,0,1,0,1,1,0,0,0,1,0] => 180 [1,1,0,1,0,1,1,0,0,1,0,0] => 180 [1,1,0,1,0,1,1,0,1,0,0,0] => 180 [1,1,0,1,0,1,1,1,0,0,0,0] => 60 [1,1,0,1,1,0,0,0,1,0,1,0] => 180 [1,1,0,1,1,0,0,0,1,1,0,0] => 90 [1,1,0,1,1,0,0,1,0,0,1,0] => 180 [1,1,0,1,1,0,0,1,0,1,0,0] => 180 [1,1,0,1,1,0,0,1,1,0,0,0] => 90 [1,1,0,1,1,0,1,0,0,0,1,0] => 180 [1,1,0,1,1,0,1,0,0,1,0,0] => 180 [1,1,0,1,1,0,1,0,1,0,0,0] => 180 [1,1,0,1,1,0,1,1,0,0,0,0] => 90 [1,1,0,1,1,1,0,0,0,0,1,0] => 60 [1,1,0,1,1,1,0,0,0,1,0,0] => 60 [1,1,0,1,1,1,0,0,1,0,0,0] => 60 [1,1,0,1,1,1,0,1,0,0,0,0] => 60 [1,1,0,1,1,1,1,0,0,0,0,0] => 15 [1,1,1,0,0,0,1,0,1,0,1,0] => 120 [1,1,1,0,0,0,1,0,1,1,0,0] => 60 [1,1,1,0,0,0,1,1,0,0,1,0] => 60 [1,1,1,0,0,0,1,1,0,1,0,0] => 60 [1,1,1,0,0,0,1,1,1,0,0,0] => 20 [1,1,1,0,0,1,0,0,1,0,1,0] => 120 [1,1,1,0,0,1,0,0,1,1,0,0] => 60 [1,1,1,0,0,1,0,1,0,0,1,0] => 120 [1,1,1,0,0,1,0,1,0,1,0,0] => 120 [1,1,1,0,0,1,0,1,1,0,0,0] => 60 [1,1,1,0,0,1,1,0,0,0,1,0] => 60 [1,1,1,0,0,1,1,0,0,1,0,0] => 60 [1,1,1,0,0,1,1,0,1,0,0,0] => 60 [1,1,1,0,0,1,1,1,0,0,0,0] => 20 [1,1,1,0,1,0,0,0,1,0,1,0] => 120 [1,1,1,0,1,0,0,0,1,1,0,0] => 60 [1,1,1,0,1,0,0,1,0,0,1,0] => 120 [1,1,1,0,1,0,0,1,0,1,0,0] => 120 [1,1,1,0,1,0,0,1,1,0,0,0] => 60 [1,1,1,0,1,0,1,0,0,0,1,0] => 120 [1,1,1,0,1,0,1,0,0,1,0,0] => 120 [1,1,1,0,1,0,1,0,1,0,0,0] => 120 [1,1,1,0,1,0,1,1,0,0,0,0] => 60 [1,1,1,0,1,1,0,0,0,0,1,0] => 60 [1,1,1,0,1,1,0,0,0,1,0,0] => 60 [1,1,1,0,1,1,0,0,1,0,0,0] => 60 [1,1,1,0,1,1,0,1,0,0,0,0] => 60 [1,1,1,0,1,1,1,0,0,0,0,0] => 20 [1,1,1,1,0,0,0,0,1,0,1,0] => 30 [1,1,1,1,0,0,0,0,1,1,0,0] => 15 [1,1,1,1,0,0,0,1,0,0,1,0] => 30 [1,1,1,1,0,0,0,1,0,1,0,0] => 30 [1,1,1,1,0,0,0,1,1,0,0,0] => 15 [1,1,1,1,0,0,1,0,0,0,1,0] => 30 [1,1,1,1,0,0,1,0,0,1,0,0] => 30 [1,1,1,1,0,0,1,0,1,0,0,0] => 30 [1,1,1,1,0,0,1,1,0,0,0,0] => 15 [1,1,1,1,0,1,0,0,0,0,1,0] => 30 [1,1,1,1,0,1,0,0,0,1,0,0] => 30 [1,1,1,1,0,1,0,0,1,0,0,0] => 30 [1,1,1,1,0,1,0,1,0,0,0,0] => 30 [1,1,1,1,0,1,1,0,0,0,0,0] => 15 [1,1,1,1,1,0,0,0,0,0,1,0] => 6 [1,1,1,1,1,0,0,0,0,1,0,0] => 6 [1,1,1,1,1,0,0,0,1,0,0,0] => 6 [1,1,1,1,1,0,0,1,0,0,0,0] => 6 [1,1,1,1,1,0,1,0,0,0,0,0] => 6 [1,1,1,1,1,1,0,0,0,0,0,0] => 1 ----------------------------------------------------------------------------- Created: Sep 27, 2011 at 19:31 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Dec 11, 2015 at 16:42 by Veronica Waite