BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Matthias Beck (San Francisco State University and Freie Universita et Berlin) DTSTART:20200924T150000Z DTEND:20200924T160000Z DTSTAMP:20260423T021153Z UID:EIMINT/3 DESCRIPTION:Title: P artitions with fixed differences between largest and smallest parts\nb y Matthias Beck (San Francisco State University and Freie Universitaet Ber lin) as part of EIMI Number Theory Seminar\n\n\nAbstract\nEnumeration resu lts on integer partitions form a classic body of mathematics going back to at least Euler\, including numerous applications throughout mathematics a nd some areas of physics. We study the number $p(n\,t)$ of partitions of $ n$ with difference $t$ between largest and smallest parts. For example\, $ p(n\,0)$ equals the number of divisors of $n$\, the function $p(n\,1)$ cou nts the nondivisors of $n$\, and $p(n\,2) = \\binom{ \\left\\lfloor \\frac n 2 \\right\\rfloor }{ 2 }$. Beyond these three cases\, the existing lite rature contains few results about $p(n\,t)$\, even though concrete evaluat ions of this partition function are featured in several entries of Sloane' s Online Encyclopedia of Integer Sequences. \n\nOur main result is an expl icit formula for the generating function $P_t(q) := \\sum_{ n \\ge 1 } p(n \,t) \\\, q^n$. Somewhat surprisingly\, $P_t(q)$ is a rational function fo r $t>1$\; equivalently\, $p(n\,t)$ is a quasipolynomial in $n$ for fixed $ t>1$ (e.g.\, the above formula for $p(n\,2)$ is an example of a quasipolyn omial with period 2). Our result generalizes to partitions with an arbitra ry number of specified distances.\n\nThis is joint work with George Andrew s (Penn State) and Neville Robbins (SF State).\n LOCATION:https://researchseminars.org/talk/EIMINT/3/ END:VEVENT END:VCALENDAR