read `ParkingStatistics.txt`:


K:=20:

print(`Explicit (Efficient!) Exact Expressions for the Centralized Moments of the Sum of Parking Functions up to the`, K, `-th moment`):
print(`in terms of the length and the Expectation`):
print(``):

print(`Definition: A (simple) parking function of length n is a permutation of a weakly increasing sequence of positive integers`):
print(seq(x[i],i=1..n)):
print(`such that for all i `):
print(` x[i]<=i `):
print(``):
print(`For any parking function, p, let X[p] be its sum of entries`):
print(``):
print(`Let E be the expectation. In this article we will present  explicit expression in terms of E and n for all centralized moments up to the `, K, `-th `):

for k from 2 to K do
gu:=collect(CentralMoment(k,E,n),E);

print(`Theorem Number`, k-1, `:  the `, k, `-th factorial moment of X is`):
print(``):
print(gu):
print(``):
print(`and in Maple format`):
print(``):
lprint(gu):
print(``):

print(`and in latex`):
print(``):
print(latex(gu)):
print(``):

od:


quit: