The $k$-elongated plane partition function modulo small powers of $5$
Russelle Guadalupe
TL;DR
The paper addresses congruences for the $k$-elongated partition diamond function $d_k(n)$ modulo powers of $5$. Employing elementary $q$-series manipulations, $5$-dissections, and the framework of $f_m$-ratios together with the auxiliary $R(q)$ and $K$ parameters, the authors derive infinite families of congruences modulo $5$, $25$, and $125$ for various $k$ and residue classes of $n$. The main contributions include new infinite families such as $d_{16}(25n+8) \equiv 0 \pmod{5}$ and $d_8(5n+1) \equiv 0 \pmod{25}$, along with more extensive results for higher indices modulo $25$ and $125$, expanding the landscape of known congruences for $d_k(n)$. These results augment existing modular-form approaches with purely elementary $q$-series techniques and illustrate how $5$-dissections can yield deep arithmetic structure in partition-type functions.
Abstract
Andrews and Paule revisited combinatorial structures known as the $k$-elongated partition diamonds, which were introduced in connection with the study of the broken $k$-diamond partitions. They found the generating function for the number $d_k(n)$ of partitions obtained by summing the links of such partition diamonds of length $n$ and discovered congruences for $d_k(n)$ using modular forms. Since then, congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary means and modular forms by many authors, most recently Banerjee and Smoot who established an infinite family of congruences for $d_5(n)$ modulo powers of $5$. We extend in this paper the list of known results for $d_k(n)$ by proving infinite families of congruences for $d_k(n)$ modulo $5,25$, and $125$ using classical $q$-series manipulations and $5$-dissections.