k -Regular partitions and overpartitions with bounded part differences
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k-Regular partitions and overpartitions with bounded part differences Bernard L. S. Lin1 · Saisai Zheng1 Received: 1 February 2020 / Accepted: 9 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials. Keywords Combinatorial proof · Difference between largest and smallest part · k-Regular partition · Overpartition · (q, z)-overGaussian polynomials Mathematics Subject Classification 05A17 · 11P84
1 Introduction A partition of a positive integer n is a weakly decreasing sequence of positive integers (λ1 , λ2 , . . . , λr ) such that λ1 + λ2 + · · · + λr = n. The λi are called the parts of the partition. The length of λ is the number of parts in λ, and the weight of λ, denoted by |λ|, is the sum of all parts. Let p(n, t) be the number of partitions of n with fixed difference t between largest and smallest part. For t > 1, Andrews et al. [3] showed that the generating function of p(n, t) satisfies
B
Bernard L. S. Lin [email protected] Saisai Zheng [email protected]
1
School of Science, Jimei University, Xiamen 361021, People’s Republic of China
123
B. L. S. Lin and S. Zheng ∞
q t−1 (1 − q) q t−1 (1 − q) − (1 − q t ) 1 − q t−1 (1 − q t ) 1 − q t−1 (q; q)t
p(n, t)q n =
n=1
+
qt 1 − q t−1
(q; q)t
.
(1.1)
Throughout the paper, we adopt the following q-series notation: (a; q)0 = 1, (a; q)n = (a; q)∞ =
n k=1 ∞
(1 − aq k−1 ), n ∈ N , (1 − aq k−1 ).
k=1
In [4], Breuer and Kronholm studied the number p(n, ˜ t) of partitions of n with difference at most t between largest and smallest parts, and proved that
p(n, ˜ t)q n =
n≥1
1 1 − qt
1 −1 . (q; q)t
(1.2)
Later, Chapman [5] provided another proof of (1.2) by using elementary q-series manipulation, involving no results deeper than the q-binomial theorem. The first author [12] also established a simple bijection to obtain the refinements of (1.2). Inspired by Chapman’s work, the first author [11] further gave a desired proof of the general result in [3, Theorem 3] as Chapman asked. In [6], Chern established the following interesting identity (1 − αq r )(1 − αq r +1 ) · · · (1 − αq r +t−2 ) r ≥1
(1 − βq r )(1 − βq r +1 ) · · · (1 − βq r +t )
qr =
q (βq − α)(1 − q t )
(α; q)t −1 , (βq; q)t
(1.3) where t is a fixed positive integer, and α, β, q are complex variables with |q| < 1, q = 0, α = βq, and (βq; q)t = 0. Let pdt (n) (resp. pot (n)) enumerate the number of partitions of n in which all parts are distinct (resp. odd) and the difference between largest and smallest parts is less than
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