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X-WR-CALNAME:Combinatorics/Partitions Seminar
X-WR-TIMEZONE:America/New_York
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DTSTART:19700308T020000
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DTSTART:19701101T020000
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20150609T111500
DTEND;TZID=America/New_York:20150609T120500
LOCATION:MB106
URL:http://www.math.psu.edu/seminars/meeting.php?id=28182
SUMMARY:Combinatorics/Partitions Seminar - MacMahon's partial fractions.
DESCRIPTION:Seminar: Combinatorics/Partitions Seminar\nTitle: MacMahon's pa
rtial fractions.\nSpeaker: Andrew Sills\, Georgia Southern University\nAbs
tract Link: http://\nAbstract: Cayley used ordinary partial fractions deco
mpositions of \n1/[(1-x)(1-x^2). . .(1-x^m)] to obtain direct formulas for
the number of\npartitions of n into at most m parts for several small val
ues of m.\nNo pattern for general m can be discerned from these\, and in p
articular the\nrational coefficients that appear in the partial fraction d
ecomposition \nbecome quite cumbersome for even moderate sized m.\n \nMacM
ahon gave a decomposition of 1/[(1-x)(1-x^2). . .(1-x^m)] into what\nhe ca
lled "partial fractions of a new and special kind" in which the\ncoefficie
nts are "easily calculable number[s]" and the sum is indexed by\nthe parti
tions of m. \n \nWhile MacMahon's derived his "new and special" partial f
ractions using \n"combinatory analysis\," the aim of this talk is to give
a preliminary report \non a fully combinatorial explanation of MacMahon's
decomposition. It seems \nlikely that this will give a combinatorial expl
anation for the coefficients\nthat appear in the ordinary partial fraction
decompositions\, which in turn\ncan be used to give a formula for the num
ber of partitions of n into \nat most m parts for arbitrary m.
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