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In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_\omega(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts…
Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which…
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $\sigma(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a…
We study functional clones, which are sets of non-negative pseudo-Boolean functions (functions $\{0,1\}^k\to\mathbb{R}_{\geq 0}$) closed under (essentially) multiplication, summation and limits. Functional clones naturally form a lattice…
The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct…
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann,…
In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…
In a previous paper, we provided a formal definition for the concept of computational irreducibility (CIR), i.e. the fact for a function f from N to N that it is impossible to compute f(n) without following approximately the same path than…
In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg. This class includes the famous identities by Ramanujan which provide a witness to the divisibility properties of $p(5n+4),$…
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give…
We give new generalizations of some q-series identities of Dilcher and Prodinger related to divisor functions. Some interesting special cases are also deduced, including an identity related to overpartitions studied by Corteel and Lovejoy.
In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for…
In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we prove that…
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the…
We use computer algebra to study polynomial identities for the trilinear operation [a,b,c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a,b,c] satisfies the alternating property in degree 3, no new…
Partitions wherein the even parts appear in two different colours are known as cubic partitions. Recently, Merca introduced and studied the function $A(n)$, which is defined as the difference between the number of cubic partitions of $n$…
Recently Amdeberhan, Sellers, and Singh introduced a new infinite family of partition functions called generalized cubic partitions. Given a positive integer $d$, they let $a_d(n)$ be the counting function for partitions of $n$ in which the…
Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined.…
This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an…
By supporting the access of multiple memory words at the same time, Bit-line Computing (BC) architectures allow the parallel execution of bit-wise operations in-memory. At the array periphery, arithmetic operations are then derived with…