arXiv subjects
Publications and source records attributed to Koustav Banerjee.
In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient $t_o(n)$ of the corresponding $q$-series is odd, then $8n+9$ is represented by the binary quadratic form $x^2+2y^2$.
In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on Zwegers' theory of indefinite theta functions together with modular transformation properties of mock theta functions.
Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.
In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta functions, illustrating the appearance of mixed modular structures in restricted partition problems. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.
Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k \leq 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5 \leq k \leq 10$. Moreover, we show that, as $k\to\infty$, $F_{k,1}(q)$ is related to Ramanujan's third order mock theta function $\omega(q)$ and to quotients of certain $q$-binomial coefficients.
We derive an asymptotic expansion with effective error bound for $u(n)$, counting the number of unimodal sequences of size $n$. We prove that $u(n)$ satisfies the higher order Tur\'{a}n inequalities for $n\geq33$ and that certain second $j$-shifted difference of $u(n)$ are positive.
Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $k\in \mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest.
Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and conjecture a second congruence family by powers of 7, which may be amenable to similar techniques.
The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. We have discovered an infinite congruence family for $d_5(n)$ modulo powers of 5. Classical methods cannot be used to prove this family of congruences. Indeed, the proof employs the recently developed localization method, and utilizes a striking internal algebraic structure which has not yet been seen in the proof of any congruence family. We believe that this discovery poses important implications on future work in partition congruences.
Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the smallest part of the partition to be $2$, then the parity bias is reversed. That is, if $q_{o}(n)$ (resp. $q_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts) where the smallest part is at least $2$, then we have $q_o(n) 7$. We also look at some more parity biases in partitions with restricted parts.
In this paper we explore Kruyswijk's method and show how to obtain congruences for cubic partition. That apart we also examine inequalities for a(n) and provide upper bound for it in the fashion of the classic partition function p(n).
Two new representations for Ramanujan's function $σ(q)$ are obtained. The proof of the first one uses the three-variable reciprocity theorem due to Soon-Yi Kang and a transformation due to R.P. Agarwal while that of the second uses the four-variable reciprocity theorem due to George Andrews and a generalization of a recent transformation of Andrews, Schultz, Yee and the second author. The advantage of these representations is that they involve free complex parameters - one in the first representation, and two in the second.