Basic hypergeometric proofs of two quadruple equidistributions of Euler-Stirling statistics on ascent sequences

24.11.2020 15:15 - 16:45

Michael Schlosser

Abstract: 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 way 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 non-terminating 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 $4\phi3$ transformation formula of base $q$, valid as an identity in a neighbourhood 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

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Ch. Krattenthaler

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