Related papers: Beyond G\"ollnitz' Theorem I: A Bijective Approach

Alladi and Gordon introduced the method of weighted words in 1993 to prove a refinement and generalisation of Schur's partition identity. Together with Andrews, they later used it to refine Capparelli's and G\"ollnitz' identities too. In…

The G\"ollnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. G\"ollnitz and B. Gordon. In this article, we present a commutative algebra proof of the G\"ollnitz-Gordon-Andrews identities. More…

We provide new Schmidt-type results through an investigation of two bijections, which are results involving partitions with parts counted only at given indices. Mork's bijection, the first of these, was originally given as a proof of…

Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects…

The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976…

Let $N\geq 1$ be squarefree with $(N,6)=1$. Let $c\phi_N(n)$ denote the number of $N$-colored generalized Frobenius partition of $n$ introduced by Andrews in 1984. We prove $$ c\phi_N(n)= \sum_{d \mid N} N/d \cdot P\left( \frac{ N}{d^2}n -…

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…

Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems. The four-colour theorem states that any map whose countries are connected can be…

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…

P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can…

We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szeg\H{o} polynomials…

Enumeration of pattern-avoiding objects is an active area of study with connections to such disparate regions of mathematics as Schubert varieties and stack-sortable sequences. Recent research in this area has brought attention to colored…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

In this paper we give an analytic proof of the identity $A_{5,3,3}(n) =B^0_{5,3,3}(n)$, where $A_{5,3,3}(n)$ counts the number of partitions of $n$ subject to certain restrictions on their parts, and $B^0_{5,3,3}(n)$ counts the number of…

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G\"ollnitz identities. Noticing the connection between their results and Boulet's earlier…

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the…

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo…

In a recent paper, Thejitha and Fathima introduced the overcolored partition function $\bar{a}_{r,s}(n)$, which enumerates overpartitions in which even parts may appear in one of $r$ colors and odd parts in one of $s$ colors, for fixed…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of…