***************************************************************************** * 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: St000859 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of parking functions of a finite Cartan type. This is given by the size of the finite torus $Q / (h+1)Q$ where $Q$ is the root lattice. This is known to be equal to $(h+1)^n$ where $n$ is the rank and $h$ is the Coxeter number. See also [1, 2] for the Weyl group action on this finite torus. ----------------------------------------------------------------------------- References: [1] Haiman, M. D. Conjectures on the quotient ring by diagonal invariants [[MathSciNet:1256101]] [2] Armstrong, D., Reiner, V., Rhoades, B. Parking spaces [[MathSciNet:3281144]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ['A',1] => 3 ['A',2] => 16 ['B',2] => 25 ['G',2] => 49 ['A',3] => 125 ['B',3] => 343 ['C',3] => 343 ['A',4] => 1296 ['B',4] => 6561 ['C',4] => 6561 ['D',4] => 2401 ['F',4] => 28561 ['A',5] => 16807 ['B',5] => 161051 ['C',5] => 161051 ['D',5] => 59049 ['A',6] => 262144 ['B',6] => 4826809 ['C',6] => 4826809 ['D',6] => 1771561 ['E',6] => 4826809 ----------------------------------------------------------------------------- Created: Jun 25, 2017 at 19:50 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 25, 2017 at 19:50 by Christian Stump