On Non-Squashing Partitions

A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the number of non-squashing partitions with distinct parts, with a specified number of parts, or with a specified maximal part. We use the results to solve a certain box-stacking problem.Comment: 15 pages, 2 fig

Variations on a result of Bressoud

The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q–series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem

An Elementary Proof of a Conjecture of Saikia on Congruences for tt--Colored Overpartitions

The starting point for this work is the family of functions $\overline{p}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by $\overline{p}_{-t}(n)$ for specific values of $t\geq 1$ have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for $\overline{p}_{-t}(n)$ for $t=5,7,11,13$. He also included the following conjecture in that paper: \newline \ %\newline \noindent Conjecture: For all $n\geq 0$ and primes $t$, we have \begin{eqnarray*} \overline{p}_{-t}(8n+1) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+2) &\equiv & 0 \pmod{4}, \\ \overline{p}_{-t}(8n+3) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+4) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+5) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+6) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+7) &\equiv & 0 \pmod{32}. \end{eqnarray*} Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia's conjecture holds for {\bf all} odd integers $t$ (not necessarily prime)

New Infinite Families of Congruences Modulo Powers of 2 for 2--Regular Partitions with Designated Summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called {\it partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd. Recently, Herden, Sepanski, Stanfill, Hammon, Henningsen, Ickes, and Ruiz proved a number of Ramanujan--like congruences for the function $PD_2(n)$ which counts the number of partitions of weight $n$ with designated summands wherein all parts must be odd. In this work, we prove some of the results conjectured by Herden, et. al. by proving the following two infinite families of congruences satisfied by $PD_2(n)$: For all $\alpha\geq 0$ and $n\geq 0,$ \begin{eqnarray*} PD_2(2^\alpha(4n+3)) &\equiv & 0 \pmod{4} \ \ \ \ \ {\textrm and} \\ PD_2(2^\alpha(8n+7)) &\equiv & 0 \pmod{8}. \end{eqnarray*} All of the proof techniques used herein are elementary, relying on classical $q$--series identities and generating function manipulations