Partition Congruences Research Articles

New congruences for broken k-diamond partitions

New congruences for broken k-diamond partitions

High order congruences for M-ary partitions

For a sequence M=(mi)i=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M=(m_{i})_{i=0}^{\\infty }$$\\end{document} of integers such that m0=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{0}=1$$\\end{document}, mi≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{i}\\ge 2$$\\end{document} for i≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i\\ge 1$$\\end{document}, let pM(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{M}(n)$$\\end{document} denote the number of partitions of n into parts of the form m0m1⋯mr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{0}m_{1}\\cdots m_{r}$$\\end{document}. In this paper we show that for every positive integer n the following congruence is true: pM(m1m2⋯mrn-1)≡0mod∏t=2rM(mt,t-1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} p_{M}(m_{1}m_{2}\\cdots m_{r}n-1)\\equiv 0\\ \\ \\left( \ extrm{mod}\\ \\prod _{t=2}^{r}\\mathcal {M}(m_{t},t-1)\\right) , \\end{aligned}$$\\end{document}where M(m,r):=mgcd(m,lcm(1,…,r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}(m,r):=\\frac{m}{\ extrm{gcd}\\big (m,\ extrm{lcm}(1,\\ldots ,r)\\big )}$$\\end{document}. Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for m-ary partitions found by Andrews, Gupta, and Rødseth and Sellers.

From sums of divisors to partition congruences

Let z be a complex number. For any positive integer n it is well known that the sum of the zth powers of the positive divisors of n can be computed without knowing all the divisors of n, if we take into account the factorization of n. In this paper, we rely on the integer partitions of n in order to investigate computational methods for ∑d|n(±1)d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (\\pm 1)^{d+1}\\,d^z$$\\end{document}, ∑d|n(-1)n/d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+1}\\,d^z$$\\end{document} and ∑d|n(-1)n/d+ddz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+d}\\,d^z$$\\end{document}. To compute these sums of divisors of n, it is sufficient to know the multiplicity of 1 in each partition involved in the computational process. Our methods do not require knowing the divisors of n or the factorization of n. New congruences involving Euler’s partition function p(n) are experimentally discovered in this context.

New infinite families of congruences for 5-core and 7-core partitions

New infinite families of congruences for 5-core and 7-core partitions

MacMahon's partition analysis XIV: Partitions with n copies of n

We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.

Modulo 2 congruences for partitions with initial repetitions

Motivated by Andrews’ partitions with initial repetitions, we derive parity formulas for several functions for this class of partitions. In many cases, we present an infinite family of Ramanujan-type congruences modulo 2.

Partition congruences and vanishing coefficients of products of theta functions

Partition congruences and vanishing coefficients of products of theta functions

A single-variable proof of the omega SPT congruence family over powers of 5

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function omega (q). Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 {mathbb {Z}}[X]-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang’s family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of {mathbb {Z}}[X]. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.

Cranks for partitions with bounded largest part

For 80 years, Dyson’s rank has been known as the partition statistic that witnesses the first two of Ramanujan’s celebrated congruences for the ordinary partition function. In this paper, we show that Dyson’s rank actually witnesses families of partition congruences modulo every prime ℓ \ell . This comes from an in-depth study of when a “multiplicity-based statistic” is a crank witnessing congruences for the function p ( n , m ) p\big (n,m\big ) , which enumerates partitions of n n with parts of size at most m m . We also show that as the modulus ℓ \ell increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same families.

Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights

Abstract We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central L {\mathrm{L}} -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo ℓ {\ell} analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo ℓ {\ell} on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work.

Proofs of two conjectural Andrews–Beck type congruences due to Lin, Peng and Toh

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of [Formula: see text] with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for [Formula: see text]-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo [Formula: see text] by utilizing some [Formula: see text]-series techniques and the theory of modular forms.

Elementary proofs of infinite families of congruences for Merca’s cubic partitions

Recently, using modular forms and Smoot’s Mathematica implementation of Radu’s algorithm for proving partition congruences, Merca proved the following two congruences: for all \(n\ge 0,\)$$\begin{aligned} A(9n+5)&\equiv 0 \pmod {3}, \\ A(27n+26)&\equiv 0 \pmod {3}. \end{aligned}$$Here, A(n) is closely related to the function which counts the number of cubic partitions, partitions wherein the even parts are allowed to appear in two different colors. Indeed, A(n) is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts. In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non-nested Ramanujan-like congruences modulo 3 satisfied by A(n) wherein Merca’s original two congruences serve as the initial members of each family.

Congruences of Hurwitz class numbers on square classes

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.

An Invitation to Formal Power Series

This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton’s binomial theorem, Jacobi’s triple product, the Rogers–Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan’s partition congruences, generating functions of Stirling numbers and Jacobi’s four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon’s master theorem.

Congruences for 3-core cubic partitions

Congruences for 3-core cubic partitions

A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}\cdot n+\dfrac{3^{2k+4}-13}{4}\right)&\equiv0~\left(\mathrm{mod}~3^{2k+5}\right).\end{align*}

Computer-assisted proofs of congruences for multipartitions and divisor function convolutions, based on methods of differential algebra

This paper provides elementary proofs for several types of congruences involving multipartitions and self-convolutions of the divisor function. Our computations use differential algebra and Gröbner bases, and employ a couple of MAPLE programs available as ancillary files on arXiv. The first results of the paper are Ramanujan-type congruences of the form \(p^{*k}(qn+r) \equiv _q 0\) and \(\sigma ^{*k}(qn+r) \equiv _q 0\), where p(n) and \(\sigma (n)\) are the partition and divisor functions, \(q > 3\) is prime, \(\equiv _q\) denotes congruence modulo q, and \(^{*k}\) denotes kth-order self-convolution. We prove all the valid congruences of this form for \(q \in \{5, 7, 11\}\), including the three famous partition congruences of Ramanujan, and an exceptional one for \(q = 17\). All such multipartition congruences have already been settled in principle (up to a numerical verification) due to Eichhorn and Ono, using modular forms. On the other hand, the majority of the divisor function congruences are new results. We then proceed to search for more general congruences modulo small primes, concerning linear combinations of \(\sigma ^{*k}(qn+r)\) for different values of k, as well as weighted convolutions of p(n) and \(\sigma (n)\) with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N.C. Bonciocat.

Proof of a Lin-Peng-Toh's conjecture on an Andrews-Beck type congruence

Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities.

Asymptotic distribution of the partition crank

Freeman Dyson conjectured the existence of an unknown partition statistic he called the crank which would explain Ramanujan’s partition congruence mod 11 just as his rank statistic explains Ramanujan’s partition congruences mod 5 and 7. Such a crank statistic was found by Andrews and Garvan in 1988. In this paper, we investigate the crank counting function, which counts the number of partitions of n with crank congruent to r mod Q. First, we obtain an effective bound on the error term in Zapata Rolón’s asymptotic formula for the crank counting function. We then use this to prove that the crank counting function is asymptotically equidistributed mod Q, for any odd number Q. We also use this to study surjectivity of the crank when viewed as a function from partitions to the integers mod Q, and to prove strict log-subadditivity of the crank counting function. The latter result is analogous to Bessenrodt and Ono’s strict log-subadditivity of the partition function.