## Seminar:

Fall 2017, Thursdays, Blocker 628, 4:00-4:50 PM

Date: October 12

"  Enumeration of Exponential Families "
R. Lorentz, S. Tringali and C. Yan introduced the sequence of generalized Goncarov Polynomials $\mathcal{T} = \{t_n(x)\}_{n\geq 0}$, which is a basis for the solutions to the Goncarov Interpolation Problem with respect to a delta operator. In the same paper, they showed that the $\{t_n(x)\}_{n\geq 0}$ satisfy the recursion $p_{n}(x)= ‎‎\sum\limits_{i=0}^{n} \binom{n}{i} p_{n-i}(z_i) t_i(x)$, where the polynomial sequence $\{p_{n}(x)\}_{n\geq 0}$ are polynomials of binomial type that count Reluctant Functions. We introduce a more general family of polynomial sequences $\{t_n(x;\mathcal{B}; Z)\}_{n\geq 0}$ associated to the enumeration of Exponential Families $\mathcal{B}$. Then, we prove that such polynomial sequences provide an algebraic tool for counting combinatorial structures with linear constraints on their order statistics.