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Using GENERATINGFUNCTIONOLOGY to Enumerate Distinct-Multiplicity Partitions
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BibTeX
@MISC{Zeilberger_usinggeneratingfunctionology,
author = {Doron Zeilberger},
title = { Using GENERATINGFUNCTIONOLOGY to Enumerate Distinct-Multiplicity Partitions},
year = {}
}
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Abstract
About a year ago, Herb Wilf[W1] posed, on-line, eight intriguing problems. I don’t know the answer to any of them, but I will say something about the sixth question. Herb Wilf’s 6th Question: Let T (n) be the set of partitions of n for which the (nonzero) multiplicities of its parts are all different, and write f(n) = |T (n)|. See Sloane’s sequence A098859 for a table of values. Find any interesting theorems about f(n)... First, I will explain how to compute the first few terms of f(n). Shalosh can easily get the first 250 terms, but as n gets larger it gets harder and harder to compute, unlike its unrestricted cousin, p(n). I conjecture that the fastest algorithm takes exponential time, but I have no idea how to prove that claim. I am impressed that, according to Sloane, Maciej Ireneusz Wilczynsk computed 508 terms. Recall that the generating function for the number of integer partitions of n whose largest part is ≤ m, pm(n), is the very simple rational function pm(n)q n = n=0 1
