Identifier
Values
[1] => 1
[1,1] => 2
[1,2] => 3
[2,1] => 3
[1,1,1] => 3
[1,1,2] => 4
[1,2,1] => 4
[2,1,1] => 4
[1,1,3] => 5
[1,3,1] => 5
[3,1,1] => 5
[1,2,2] => 5
[2,1,2] => 5
[2,2,1] => 5
[1,2,3] => 6
[1,3,2] => 6
[2,1,3] => 6
[2,3,1] => 6
[3,1,2] => 6
[3,2,1] => 6
[1,1,1,1] => 4
[1,1,1,2] => 5
[1,1,2,1] => 5
[1,2,1,1] => 5
[2,1,1,1] => 5
[1,1,1,3] => 6
[1,1,3,1] => 6
[1,3,1,1] => 6
[3,1,1,1] => 6
[1,1,1,4] => 7
[1,1,4,1] => 7
[1,4,1,1] => 7
[4,1,1,1] => 7
[1,1,2,2] => 6
[1,2,1,2] => 6
[1,2,2,1] => 6
[2,1,1,2] => 6
[2,1,2,1] => 6
[2,2,1,1] => 6
[1,1,2,3] => 7
[1,1,3,2] => 7
[1,2,1,3] => 7
[1,2,3,1] => 7
[1,3,1,2] => 7
[1,3,2,1] => 7
[2,1,1,3] => 7
[2,1,3,1] => 7
[2,3,1,1] => 7
[3,1,1,2] => 7
[3,1,2,1] => 7
[3,2,1,1] => 7
[1,1,2,4] => 8
[1,1,4,2] => 8
[1,2,1,4] => 8
[1,2,4,1] => 8
[1,4,1,2] => 8
[1,4,2,1] => 8
[2,1,1,4] => 8
[2,1,4,1] => 8
[2,4,1,1] => 8
[4,1,1,2] => 8
[4,1,2,1] => 8
[4,2,1,1] => 8
[1,1,3,3] => 8
[1,3,1,3] => 8
[1,3,3,1] => 8
[3,1,1,3] => 8
[3,1,3,1] => 8
[3,3,1,1] => 8
[1,1,3,4] => 9
[1,1,4,3] => 9
[1,3,1,4] => 9
[1,3,4,1] => 9
[1,4,1,3] => 9
[1,4,3,1] => 9
[3,1,1,4] => 9
[3,1,4,1] => 9
[3,4,1,1] => 9
[4,1,1,3] => 9
[4,1,3,1] => 9
[4,3,1,1] => 9
[1,2,2,2] => 7
[2,1,2,2] => 7
[2,2,1,2] => 7
[2,2,2,1] => 7
[1,2,2,3] => 8
[1,2,3,2] => 8
[1,3,2,2] => 8
[2,1,2,3] => 8
[2,1,3,2] => 8
[2,2,1,3] => 8
[2,2,3,1] => 8
[2,3,1,2] => 8
[2,3,2,1] => 8
[3,1,2,2] => 8
[3,2,1,2] => 8
[3,2,2,1] => 8
[1,2,2,4] => 9
[1,2,4,2] => 9
[1,4,2,2] => 9
[2,1,2,4] => 9
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Description
The sum of the entries of a parking function.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
References
[1] Merino López, C. Chip firing and the Tutte polynomial MathSciNet:1630779
Code
def statistic(p):
return sum(p)
def generating_function(n):
R. = ZZ[]
return graphs.CompleteGraph(n).tutte_polynomial().subs(x=1, y=1/q)*q^binomial(n, 2)
Created
Nov 05, 2013 at 18:44 by Jeremy L. Martin
Updated
Mar 27, 2024 at 13:49 by Martin Rubey
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