# Meeting Details

Title: Properties of a Restricted Binary Partition Function a la Andrews and Lewis Combinatorics/Partitions Seminar James Sellers, Penn State In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. At the end of their paper, they define a family of functions $W_1(S_1, S_2;n)$ to be the number of partitions of $n$ into parts from $S_1 \cup S_2$ which do not contain both $a_j$ and $b_j$ as parts (where $S_1 = \left\{ a_1, a_2, a_3, \dots\right\}$ and $S_2 = \left\{ b_1, b_2, b_3, \dots\right\}$ and $S_1 \cap S_2 = \phi$). This definition is motivated by the main results of their paper; in that case, $S_1$ and $S_2$ contain elements in arithmetic progression with the same skip value'' $k$. Our goal in this work is to consider more general examples of such partition functions where $S_1$ and $S_2$ satisfy the requirements mentioned above but do not simply contain elements in an arithmetic progression. In particular, we consider the situation where $S_1$ and $S_2$ contain specific powers of $2.$ We then prove a number of arithmetic properties satisfied by this function using elementary generating function manipulations and classic results from the theory of partitions. This work was completed in collaboration with my undergraduate student Bin Lan.