The next talk in the seminar is by Michael Schlosser. This talk is a longer version of a talk he presented recently in an AMS regional meeting. Please note that this time the talk will be on Zoom.
Talk announcement
Title: Basic hypergeometric proofs of two quadruple equidistributions of Euler-Stirling statistics on ascent sequences
Speaker: Michael Schlosser (University of Vienna, Austria).
When: Thursday October 22, 3:55 PM-5:00 PM
Where: ZOOM this time. Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance. (The link will be open at 3:30 PM for the organizers to test their systems)
Tea or Coffee: Please bring your own.
Abstract:
In my talk, I will present new applications of basic hypergeometric series to specific problems in enumerative combinatorics. The combinatorial problems we are interested in concern multiply refined equidistributions on ascent sequences. (I will gently explain these notions in my talk!) Using bijections we are able to suitably decompose some quadruple distributions we are interested in and obtain functional equations and ultimately generating functions from them, in the form of explicit basic hypergeometric series. The problem of proving equidistributions then reduces to applying suitable transformations of basic hypergeometric series. The situation in our case however is tricky (caused by the fact how the power series variable $r$ appears in the base $q=1-r$ of the respective basic hypergeometric series; so being interested in the generating function in $r$ as a Maclaurin series, we are thus interested in the analytic expansion of the nonterminating basic hypergeometric series in base $q$ around the point $q=1$), as none of the known transformations appear to directly work to settle our problems; we require the derivation of new identities. Specifically, we use the classical Sears transformation and apply some analytic tools to establish a new non terminating ${}_4\phi_3$ transformation formula of base $q$, valid as an identity in a neighborhood around $q=1$. We use special cases of this formula to deduce two different quadruple equidistribution results involving Euler--Stirling statistics on ascent sequences. One of them concerns a symmetric equidistribution, the other confirms a bi symmetric equidistribution that was recently conjectured in a paper (published in JCTA) by Shishuo Fu, Emma Yu Jin, Zhicong Lin, Sherry H.F. Yan, and Robin D.B. Zhou.
This is joint work with Emma Yu Jin. For full results (and further ones), see https://arxiv.org/abs/2010.014
