arXiv subjects
Publications and source records attributed to Shane Chern.
We study the asymptotics for the Fourier coefficients of a broad class of eta-quotients, $$\prod_{r=1}^R \left(\prod_{k\ge 1}\left(1-q^{m_r k}\right)\right)^{δ_r},$$ where $m_1,\ldots,m_R$ are $R$ distinct positive integers and $δ_1,\ldots,δ_R$ are $R$ non-zero integers with $\sum_{r=1}^R δ_{r}\ge 0$.
Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_ω(n)$ and $p_ν(n)$ introduced by Andrews, Dixit and Yee. In this paper, we present a combinatorial proof of an interesting identity in their work.
We generalize recent results of Breuer and Kronholm, and Chern on partitions and overpartitions with bounded differences between largest and smallest parts. We prove our generalization both analytically and combinatorially.
In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.
In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis' results on the rank and crank counts for ordinary partitions.
Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy's identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.
In this note, we investigate J.-C. Liu's work on truncated Gauss' square exponent theorem and obtain more truncations. We also discuss some possible multiple summation extensions of Liu's results.
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 $β(q)=\sum_{n\ge 0} \mathfrak{b}(n)q^n$ be a second order mock theta function defined by $$\sum_{n\ge 0}\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)_{n+1}^2}.$$ In this paper, we obtain an infinite family of congruences modulo powers of $3$ for $\mathfrak{b}(n)$.
We introduce a new type of partitions that consists of partitions whose different parts alternate in parity (e.g., $3+2+2+1+1$). Various properties of this partition function are studied. In particular, we obtain its asymptotic behavior by employing Ingham's Tauberian theorem.
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
In this paper, we study the power mean of a sum analogous to the Kloosterman sum by using analytic methods for character sums.
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.
Recently, Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this paper, we provide a combinatorial proof of the generating function identity of such partitions. We also have a further investigation on the largest even part. Finally, we give an interesting weighted overpartition generalization.
Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the third and sixth order mock theta functions. In addition, three Ramanujan-type congruences were established by them. In this paper, we present some new congruences for these partition functions.
For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this note, we shall give an explicit bound of $|\mathcal{R}_n^*(\mathcal{S})|$. We also present an application of this bound to reducible bivariate polynomials over $\mathbb{Z}[X,Y]$.
Let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of weight $n$. Recently, Hirschhorn and Sellers, Yao, and Xia established a number of congruences modulo 2 and 5, 4 and 8, and 25 for $f(n)$, respectively. In this note, we shall prove several new congruences modulo 125 and 11 by using some results of modular forms. For example, for all $n\ge 0$, we have \begin{align*} f(1250n+125)&\equiv 0 \pmod{125},\\ f(1250n+1125)&\equiv 0 \pmod{125},\\ f(2750n+825)&\equiv 0 \pmod{11},\\ f(2750n+1925)&\equiv 0 \pmod{11}. \end{align*}
Recently, Shen (2016) and Alanazi et al. (2016) studied the arithmetic properties of the $\ell$-regular overpartition function $\overline{A}_\ell (n)$, which counts the number of overpartitions of $n$ into parts not divisible by $\ell$. In this note, we will present some new congruences modulo $5$ when $\ell$ is a power of $5$.