# Brandt Kronholm

Affiliation: University Of Texas Rio Grande Valley

Email: brandt.kronholm@utrgv.edu

Title Of Talk: Quasipolynomials, Polyhedral Geometry, Divisibility Patterns and Combinatorial Witnesses for Partitions

Abstract: In this presentation, we discuss the quasipolynomial for the restricted partition function $p(n,m)$ which enumerates the number of partitions of $n$ into exactly $m$ parts. We show that it treats both divisibility properties of these restricted partition numbers but also encodes a combinatorial witness to these divisibilities.

\begin{eqnarray*} p(6k,3) & = 0{k+2 \choose 2} + 3{k+1 \choose 2} + 3{k \choose 2}& ~\\ p(6k+1,3) & = 0{k+2 \choose 2} + 4{k+1 \choose 2} + 2{k \choose 2}& ~\\ p(6k+2,3) & = 0{k+2 \choose 2} + 5{k+1 \choose 2} + 1{k \choose 2}& ~\\ p(6k+3,3) & = 1{k+2 \choose 2} + 4{k+1 \choose 2} + 1{k \choose 2} & ~\\ p(6k+4,3) & = 1{k+2 \choose 2} + 5{k+1 \choose 2} + 0{k \choose 2} & ~\\ p(6k+5,3) & = 2{k+2 \choose 2} + 4{k+1 \choose 2} + 0{k \choose 2} & \end{eqnarray*}

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