In 2003, Hammond and Lewis defined a statistic on partitions into 2 colors which combinatorially explains certain well known partition congruences mod 5. We give two analogs of Hammond and Lewis's birank statistic. One analog is in terms of Dyson's rank and the second uses the 5-core crank due to Garvan, Kim and Stanton. We discuss Andrews's bicrank statistic and how it may be extended. We also generalize the Hammond-Lewis birank to a multirank for multipartitions and the Andrews bicrank to a multicrank for extended multipartitions. These both give combinatorial interpretations for multipartition congruences modulo all primes t>3.
In this paper, motivated by the work of Mahlburg, we find congruences for a large class of modular forms. Moreover, we generalize the generating function of the Andrews-Garvan-Dyson crank on partition and establish several new infinite families of congruences. In this framework, we showed that both the birank of an ordered pair of partitions introduced by Hammond and Lewis, and $k$-crank of $k$-colored partition introduced by Fu and Tang process the same as the partition function and crank.
We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.
Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
Motivated by a question of Lovejoy \cite{lovejoy}, we show that three-colored Frobenius partition function $\c3$ and related arithmetic fuction $\cc3$ vanish modulo some powers of 5 in certain arithmetic progressions.
Let ${p}_{3,3}(n)$ denote the number of $2$-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$. In this paper, we shall establish some interesting Ramanujan-type congruences for ${p}_{3,3}(n)$.
Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$. Moreover, we will present some new congruences modulo $7$ for $p_4(n)$.
In $2003$, Alladi, Andrews and Berkovich proved a four-parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity. In this second paper, we state and bijectively prove an $\fra…
Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n := \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that for every non-negative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv 0\pmod{5}. As a corollary, we obtained some results of Ono
Let $cϕ_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $cϕ_{3}(n)$ and $cϕ_{9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that \[cϕ_{3}\Big(3^{2k}n+\frac{7\cdot 3^{2k}+1}{8}\Big) \equiv 0 \pmod{3^{4k+5}}.\] We give two different proofs to the congruences satisfied by $cϕ_{9}(n)$. One of the proofs uses an relation between $cϕ_{9}(n)$ and $cϕ_{3}(n)$ due to Kolitsch, for which we provide a new proof in this paper.
We investigate congruence relations of the form $p_r(\ell m n + t) \equiv 0 \pmod{\ell}$ for all $n$, where $p_r(n)$ is the number of $r$-colored partitions of $n$ and $m,\ell$ are distinct primes.
Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS's), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of "ordered" triple systems (viz. those with a certa…
Let $k\in\N_{\geq 2}$ and for given $m\in\Z\setminus\{0\}$ consider the sequence $(S_{k,m}(n))_{n\in\N}$ defined by the power series expansion $$ \frac{1}{(1-x)^{m}}\prod_{i=0}^{\infty}\frac{1}{(1-x^{k^{i}})^{m}}=\sum_{n=0}^{\infty}S_{k,m}(n)x^{n}. $$ The number $S_{k,m}(n)$ for $m\in\N_{+}$ has a natural combinatorial interpretation: it counts the number of representations of $n$ as sums of powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each part $>1$ takes $m(k-1)$ colors. We concentrate on the case when $k=p\in\mathbb{P}$. Our main result is the computation of th…
The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples of $k$. In this paper, we record the complete characterization of the modulo 5 congruence relation $p_{k}(25n + 24 - k) \equiv 0 \pmod{5}$ for $k \in \{1, 2, \ldots, 24\}$, in connection with the 2-color partition function $p_k(n)$, providing references to existing results for $k \in \{1, 2, 3, 4, 7, 8, 17\}$, simple proofs for $k \in \{5, 10, 15, 20\}$ for …
In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $cϕ_k(n)$ continues, with recent works completed b…
Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+β) \equiv 0 \pmod{\ell}.$ Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$. In this paper, for a wide range of $c \in \mathbb{F}_{\ell}$, we prove congruences of the form $p(\ell Q^{3} n+β_{0}) \equiv c \cdot p(\ell Q n+β_{1}) \pmod{\ell}$ for infinitely many primes $Q$. For a positive integer $r$, let $p_{r}(n)$ be the $r$-colored par…
This note rederives a formula for $r$-color partitions, $1 \le r \le 24$, including Rademacher's celebrated result for ordinary partitions, from the duality between modular forms of weights $-r/2$ and $2+r/2$.
