Using Symbolic Moment Calculus (and Experimental Mathematics) to Study the Area Statistics of Parking Functions
By Yukun Yao and Doron Zeilberger
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First Written: May ?, 2018
Maple package
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ParkingStatistics.txt,
a Maple package for automatically deriving explicit expressions for moments of the random variable "Sum of Entries"
defined on parking functions, and for computing the limiting scaled distribution.
Pictures produced by ParkingStatistics.txt
Sample Input and Output for ParkingStatistics.txt
-
If you want to see explicit polynomial expressions for the first thirty factorial moments of the random variable
"area" (i.e. n(n+1)/2-sum of entries) defined on the "sample space" of all (n+1)^(n-1) parking functions of
length n, as polynomial expressions in n and the expectation A1 (that equals [in Maple notation])
1/2*(n+1)!/(n+1)^n*add((n+1)^k/k!,k=0..n-1) - n/2
the input file generates the
output file.
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If you want to see the first 70 terms of the sequence of weight-enumerators for parking functions, according to the sum,
and the statistical information for those between 60 and 70,
the input file generates the
output file.
-
If you want to statistical information about the random variable "sum of the parking function"
and the statistical information for those between 121 and 130,
the input file generates the
output file.
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If you want to see EXPLICIT expressions for the second through 20th FACTORIAL moments
of the random variable "sum of the parking function" , for (simple) parking functions of length n, in terms of
n and the Expectation
the input file generates the
output file.
-
If you want to see EXPLICIT expressions for the second through 20th CENTRALIZED moments
of the random variable "sum of the parking function" , for (simple) parking functions of length n, in terms of
n and the Expectation
the input file generates the
output file.
-
If you want to see EXPLICIT expressions for the second through 20th Limits (as n goes to infinity) of the SCALED moments
of the random variable "sum of the parking function" , for (simple) parking functions of length n, in terms of
n and the Expectation
the input file generates the
output file.
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If you want to see EXPLICIT expressions for the second through 10th FACTORIAL moments
of the random variable "sum of the parking function" , for a-parking functions of length n, in terms of
n, a and the Expectation
the input file generates the
output file.
Yukon Yao's Home Page
Articles of Doron Zeilberger
Doron Zeilberger's Home Page