Where: Zoom: Please write to sf-and-nt@gmail.com for the link.

Tea or Coffee: Please bring your own.

Abstract:We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their inverses. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $R(q)=\left(q,q^4;q^5\right)_{\infty}/\left(q^2,q^3;q^5\right)_{\infty}$. If \begin{align*} \sum_{n=0}^{\infty}\alpha(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}\alpha^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}\beta(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}\beta^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} \alpha(5n+r)&=-\alpha^{\prime}(5n+r-2), \quad r\in\{3,4\}; \\ \text{and} & \\ \beta(10n+r)&=-\beta^{\prime}(10n+r-6), \quad r\in\{7,9\}. \end{align*} This is a joint work with Hirakjyoti Das.

The next speaker in our seminar is Shishuo Fu of Chongqing University, PRC. It may be Fool's day, but we're not kidding. It really is Shishuo who has consented to give a talk all the way from China!

The live broadcast did not work as anticipated in the previous talk; I hope it works this time. At any rate, its best to try and come for the zoom session.

Talk Announcement

Title: Bijective recurrences for Schroeder triangles and Comtet statistics

Speaker: Shishuo Fu (Chongqing University, PRC)

When: April 1, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom: Please write to sfandnt@gmail.com for a link

Tea or Coffee: Please bring your own.

Abstract:

In this talk, we bijectively establish recurrence relations for two triangular arrays, relying on their interpretations in terms of Schroeder paths (resp. little Schroeder paths) with given length and number of hills. The row sums of these two triangles produce the large (resp. little) Schroeder numbers. On the other hand, it is well-known that the large Schroeder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run (iar), whose distribution on separable permutations is shown to be given by the first triangle as well. A by-product of this result is that "iar" is equidistributed over separable permutations with "comp", the number of components of a permutation. We call such statistics Comtet and we briefly mention further work concerning Comtet statistics on various classes of pattern avoiding permutations. The talk is based on joint work with Zhicong Lin and Yaling Wang.