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Statistic identifier: St000859

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Collection: Finite Cartan types

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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.

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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]]

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Code:


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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

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Created: Jun 25, 2017 at 19:50 by Christian Stump

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Last Updated: Jun 25, 2017 at 19:50 by Christian Stump