Related papers: Linear identities for partition pairs with $4$-cor…

In this note, we will give a short proof of an identity for cubic partitions.

Andrews and El Bachraoui recently studied integer partitions where the smallest part is repeated a specified number of times and any other parts are distinct. Their results included two ``surprising identities'' for which they requested…

Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and…

Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made…

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…

Four dimensional $\mathcal{N}=2$ Argyres-Douglas theories have been recently conjectured to be described by $\mathcal{N}=1$ Lagrangian theories. Such models, once reduced to 3d, should be mirror dual to Lagrangian $\mathcal{N}=4$ theories.…

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is…

Euler's partition identity states that the number of partitions of $n$ into odd parts is equal to the number of partitions of $n$ into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions…

First we prove some elementary but useful identities in the group ring of Q/Z. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together…

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence…

In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee…

Picone-type identities are established for half-linear ODEs of fourth order (one-dimensional p-biLaplacian). It is shown that in the linear case they reduce to the known identities for fourth order linear ODEs. Picone-type identity known…

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,…

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left(…

We extend the theory of infinite-exponent partition relations to arbitrary linear order types, with a particular focus on the real number line. We give a complete classification of all consistent partition relations on the real line with…

We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, $m$-separability and entanglement depth are detected as…

We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We…

Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2\alpha}n+\frac{3^{2\alpha}-1}{4}\right…

Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type…

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including…