An overpartition analogue of partitions with bounded differences between largest and smallest parts

TL;DR

This paper investigates the generating function for overpartitions with bounded differences between largest and smallest parts, linking it to over q-binomial coefficients and extending classical partition results.

Contribution

It introduces an overpartition analogue of a classical partition problem and connects it with over q-binomial coefficients, providing new insights into overpartition structures.

Findings

Derived the generating function for overpartitions with bounded differences.

Connected overpartition problems with over q-binomial coefficients.

Extended classical partition results to the overpartition context.

Abstract

We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer partitions. We also connect this problem with over $q$-binomial coefficients introduced by Dousse and Kim.

Equations

$$1+\sum _{n\ge 1}p(n)q^n=\frac{1}{(q{)}_{\infty }},$$

$$(a{)}_n=(a;q{)}_n:=\prod _{k=0}^{n-1}(1-a{q}^k),$$

$$(a{)}_{\infty }=(a;q{)}_{\infty }:=\underset{n\to \infty }{\lim }(a{)}_n.$$

$$\sum _{n\ge 1}p(n,t)q^n=\frac{q^{t-1}(1-q)}{(1-q^t)(1-q^{t-1})}-\frac{q^{t-1}(1-q)}{(1-q^t)(1-q^{t-1})(q{)}_t}+\frac{q^t}{(1-q^{t-1})(q{)}_t}.$$

$$\sum _{n\ge 1}{p}_t(n)q^n=\frac{1}{1-q^t}\left(\frac{1}{(q{)}_t}-1\right).$$

$$4,\overline{4},3+1,\overline{3}+1,3+\overline{1},\overline{3}+\overline{1},2+2,\overline{2}+2,$$

$$2+1+1,\overline{2}+1+1,2+\overline{1}+1,\overline{2}+\overline{1}+1,1+1+1+1,\overline{1}+1+1+1.$$

$$1+\sum _{n\ge 1}\overline{p}(n)q^n=\frac{(-q{)}_{\infty }}{(q{)}_{\infty }}.$$

$$4,\overline{4},2+2,\overline{2}+2,2+1+1,\overline{2}+1+1,$$

$$2+\overline{1}+1,\overline{2}+\overline{1}+1,1+1+1+1,\overline{1}+1+1+1,$$

$$4,\overline{4},2+2,\overline{2}+2,2+1+1,$$

$$2+\overline{1}+1,1+1+1+1,\overline{1}+1+1+1.$$

$${\overline{P}}_t(q):=\sum _{n\ge 1}{\overline{p}}_t(n)q^n\text{and}{G}_t(q):=\sum _{n\ge 1}{g}_t(n)q^n.$$

$$G_t(q)=\frac{1}{1-q^t}\left(\frac{(-q{)}_t}{(q{)}_t}-1\right).$$

$${\overline{P}}_t(q)=2(-1{)}^t\left(\sum _{n\ge 1}d(n)q^n+\sum _{n=1}^t\frac{(-1{)}^n}{1-q^n}\left(\frac{(-q{)}_n}{(q{)}_n}-1\right)\right).$$

$${}_{r+1}{\varphi }_s\left(\begin{array}{c}{a}_0,a_1,a_2\dots ,a_r\\ b_1,b_2,\dots ,b_s\end{array};q,z\right):=\sum _{n\ge 0}\frac{(a_0;q{)}_n(a_1;q{)}_n\cdots (a_r;q{)}_n}{(q;q{)}_n(b_1;q{)}_n\cdots (b_s;q{)}_n}{\left((-1{)}^n{q}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\right)}^{s-r}{z}^n.$$

$${}_2{\varphi }_1\left(\begin{array}{c}a,q^{-n}\\ c\end{array};q,c{q}^n/a\right)=\frac{(c/a;q{)}_n}{(c;q{)}_n}.$$

$${}_3{\varphi }_2\left(\begin{array}{c}a,b,c\\ d,e\end{array};q,de/(abc)\right)=\frac{(e/a;q{)}_n(de/(bc);q{)}_n}{(e;q{)}_n(de/(abc);q{)}_n}{}_3{\varphi }_2\left(\begin{array}{c}a,d/b,d/c\\ d,de/(bc)\end{array};q,e/a\right).$$

$$\frac{2q^m}{1-q^m}\frac{1+q^{m+1}}{1-q^{m+1}}\cdots \frac{1+q^{m+t-1}}{1-q^{m+t-1}}\frac{1}{1-q^{m+t}}.$$

$$\frac{G_t(q)}{2}$$

$$=\sum _{m\ge 1}\frac{(q{)}_{m-1}(-q{)}_{m+t-1}}{(q{)}_{m+t}(-q{)}_m}{q}^m$$

$$=q\sum _{m\ge 0}\frac{(q{)}_m(-q{)}_{m+t}}{(q{)}_{m+t+1}(-q{)}_{m+1}}{q}^m$$

$$=\frac{q(-q{)}_t}{(1+q)(q{)}_{t+1}}\sum _{m\ge 0}\frac{(q{)}_m(q{)}_m(-q^{t+1}{)}_m}{(q{)}_m(q^{t+2}{)}_m(-q^2{)}_m}{q}^m$$

$$=\frac{q(-q{)}_t}{(1+q)(q{)}_{t+1}}{}_3{\varphi }_2\left(\begin{array}{c}q,q,-q^{t+1}\\ -q^2,q^{t+2}\end{array};q,q\right)$$

$$=\frac{q(-q{)}_t}{(1+q)(q{)}_{t+1}}\frac{(q^{t+1}{)}_{\infty }(q^2{)}_{\infty }}{(q^{t+2}{)}_{\infty }(q{)}_{\infty }}{}_3{\varphi }_2\left(\begin{array}{c}q,-q,q^{1-t}\\ -q^2,q^2\end{array};q,q^{t+1}\right)$$

$$=\frac{q(-q{)}_t}{(1-q^2)(q{)}_t}\sum _{m\ge 0}\frac{(-q{)}_m(q^{1-t}{)}_m}{(-q^2{)}_m(q^2{)}_m}{q}^{m(t+1)}$$

$$=\frac{q(-q{)}_t}{(1-q^2)(q{)}_t}\sum _{m\ge 0}\frac{\frac{(-1{)}_{m+1}}{2}\frac{(q^{-t}{)}_{m+1}}{1-q^{-t}}}{\frac{(-q{)}_{m+1}}{1+q}\frac{(q{)}_{m+1}}{1-q}}{q}^{(m+1)(t+1)}{q}^{-(t+1)}$$

$$=-\frac{(-q{)}_t}{2(1-q^t)(q{)}_t}\left({}_2{\varphi }_1\left(\begin{array}{c}-1,q^{-t}\\ -q\end{array};q,q^{t+1}\right)-1\right)$$

$$=-\frac{(-q{)}_t}{2(1-q^t)(q{)}_t}\left(\frac{(q{)}_t}{(-q{)}_t}-1\right)$$

$$=\frac{1}{2(1-q^t)}\left(\frac{(-q{)}_t}{(q{)}_t}-1\right).$$

$$G_t(q)=\frac{1}{1-q^t}\left(\frac{(-q{)}_t}{(q{)}_t}-1\right).$$

$${\overline{P}}_0(q)=2\sum _{n\ge 1}d(n)q^n,$$

$${\overline{P}}_t(q)=2\sum _{m\ge 1}\frac{q^m}{1-q^m}\frac{1+q^{m+1}}{1-q^{m+1}}\cdots \frac{1+q^{m+t}}{1-q^{m+t}}.$$

$${\overline{P}}_n(q)+{\overline{P}}_{n-1}(q)$$

$$=4\sum _{m\ge 1}\frac{q^m}{1-q^m}\frac{1+q^{m+1}}{1-q^{m+1}}\cdots \frac{1+q^{m+n-1}}{1-q^{m+n-1}}\frac{1}{1-q^{m+n}}$$

$$=2G_n(q).$$

Full text

Keywords. Overpartition, bounded difference between largest and smallest parts, over $q$-binomial coefficient.

2010MSC. Primary 05A17; Secondary 11P84.

1. Introduction

A partition of a positive integer $n$ is a non-increasing sequence of positive integers $(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ with $n=\lambda_{1}+\lambda_{2}+\ldots+\lambda_{k}$. For example, $4$ has five partitions: $4$, $3+1$, $2+2$, $2+1+1$, $1+1+1+1$. Let $p(n)$ denote the number of partitions of $n$. It is well known that its generating function is

[TABLE]

where, as usual, we use the standard notations

[TABLE]

and

[TABLE]

Recently, Andrews, Beck, and Robbins [2] considered partitions where the difference between largest and smallest parts is a fixed integer $t$. Let $p(n,t)$ be the number of such partitions of $n$. We have, for example, $p(4,1)=1$ since $4$ has only one such partition: $2+1+1$. In fact, Andrews et al. showed that $p(n,0)=d(n)$ and $p(n,1)=n-d(n)$ where, and in the sequel, $d(n)$ denotes the number of divisors of $n$. For $t\geq 2$, they obtained the following generating function

[TABLE]

In a subsequent paper [4], Breuer and Kronholm further considered partitions where the difference between largest and smallest parts is at most $t$. When $t=1$, the four such partitions of $4$ are: $4$, $2+2$, $2+1+1$, $1+1+1+1$. Let $p_{t}(n)$ be the number of such partitions of $n$. Its generating function is, in fact, even neater. Breuer and Kronholm proved in two ways (one is geometric and the other is combinatorial) that for $t\geq 1$,

[TABLE]

Later on, Chapman [5] also provided an analytic proof which only uses elementary $q$-series manipulation as deep as the $q$-binomial theorem.

In this paper, we shall study an overpartition analogue of Breuer and Kronholm’s result.

2. An overpartition analogue

An overpartition of $n$ is a partition of $n$ where the first occurrence of each distinct part may be overlined. For example, $4$ has fourteen overpartitions:

[TABLE]

[TABLE]

Overpartitions have many applications in combinatorics [6], mathematical physics [8], representation theory [9], and symmetric functions [3]. We denote by $\overline{p}(n)$ the number of overpartitions of $n$. It is known that

[TABLE]

Given a non-negative integer $t$, let $\overline{p}_{t}(n)$ denote the number of overpartitions of $n$ whose difference between largest and smallest parts is at most $t$. Furthermore, let $g_{t}(n)$ be the number of such overpartitions of $n$ with one more restriction: if the difference between largest and smallest parts is exactly $t$, then the largest parts cannot be overlined. For example, we have $\overline{p}_{1}(4)=10$:

[TABLE]

[TABLE]

and $g_{1}(4)=8$:

[TABLE]

[TABLE]

It is clear that $\overline{p}_{0}(n)=2d(n)$ and $g_{0}(n)=d(n)$. We now write

[TABLE]

Theorem 2.1**.**

For $t\geq 1$, we have

[TABLE]

Remark 2.1**.**

It is of interest to compare the similarity between this generating function and the result of Breuer and Kronholm (Eq. (1.2)).

Theorem 2.2**.**

For $t\geq 0$, we have

[TABLE]

Before presenting our proofs, we need several $q$-series identities. At first, we introduce the ${}_{r}\phi_{s}$ functions:

[TABLE]

Lemma 2.3** (First $q$-Chu–Vandermonde Sum [1, Eq. (17.6.2)]).**

We have

[TABLE]

Lemma 2.4** ([1, Eq. (17.9.6)]).**

We have

[TABLE]

Proof of Theorem 2.1.

Fix the smallest part to be $m\geq 1$. It is easy to see that, for $t\geq 1$, the generating function for overpartitions, where the smallest part is $m$, the largest part is at most $m+t$, and if the largest part is exactly $m+t$, then the largest parts cannot be overlined, is

[TABLE]

Hence,

[TABLE]

We therefore have

[TABLE]

This ends the proof of Theorem 2.1. ∎

Proof of Theorem 2.2.

We first note that

[TABLE]

and for $t\geq 0$,

[TABLE]

We also have, for $n\geq 1$,

[TABLE]

Hence, for any $t\geq 0$, it follows that

[TABLE]

∎

Corollary 2.5**.**

For any $t\geq 0$ and $n\geq 1$, $\overline{p}_{t}(n)$ is an even integer. Furthermore, $\overline{p}_{t}(n)$ is divisible by $4$ if and only if $n$ is not a perfect square.

Proof.

From the previous proof, we see that

[TABLE]

It follows that

[TABLE]

We finally note that $d(n)$ is odd if and only if $n$ is a perfect square. This completes the proof. ∎

3. The viewpoint of over $q$-binomial coefficients

The $q$-binomial coefficient, also known as Gaussian polynomial, is defined as

[TABLE]

We know that it is the generating function for partitions where the largest part is at most $M$ and the number of parts is at most $N$. In a recent paper [7], Dousse and Kim introduced the over $q$-binomial coefficient, denoted by

[TABLE]

which is an overpartition analogue of $q$-binomial coefficient defined as the generating function for overpartitions where the largest part is at most $M$ and the number of parts is at most $N$. They showed that for positive integers $M$ and $N$

[TABLE]

Of course, if we agree that the number of such overpartitions of [math] is one, then this identity also holds for $M=0$ or $N=0$.

Over $q$-binomial coefficients have many properties similar to those of the standard $q$-binomial coefficients. For example, the following recurrence relation

[TABLE]

holds for any positive integers $M$ and $N$ (see [7, Eq. (1.1)]). In fact, it can be proved combinatorially.

Motivated by [5], we provide an alternative proof of Theorem 2.1 using over $q$-binomial coefficients. It avoids the application of complicated $q$-series identities such as the $q$-Chu–Vandermonde sum.

Second proof of Theorem 2.1.

Here we always assume $t$ to be a positive integer. Let $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ be an overpartition of $n$ with exactly $r$ parts, $\lambda_{r}=m\geq 1$, and $\lambda_{1}\leq m+t$. Then $\mu=(\lambda_{1}-m,\ldots,\lambda_{r-1}-m)$ is an overpartition of $n-rm$ with at most $r-1$ parts and greatest part $\leq t$. Note that the first occurrence of the smallest part of $\lambda$ can be either overlined or not. Hence the generating function for such overpartitions is

[TABLE]

and hence

[TABLE]

We remark that this identity also holds for $t=0$.

On the other hand, overpartitions where the difference between largest and smallest parts is at most $t$ can be divided into three disjoint cases:

1. (1)

The largest part is at most $t$; 2. (2)

The largest part is greater than $t$, the difference between largest and smallest parts is exactly $t$, and the first occurrence of the smallest part is overlined; 3. (3)

Otherwise.

For Case (1), one readily sees the generating function is

[TABLE]

For Case (2), its generating function is

[TABLE]

Finally, let $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ be an overpartition of Case (3) with $\lambda_{1}=m+t$ (and so $m\geq 1$). We note that $\mu=(\lambda_{1}-m,\ldots,\lambda_{r}-m)$ is an overpartition of $|\lambda|-rm$ with at most $r$ parts and largest part being exactly $t$. Hence the generating function is

[TABLE]

We therefore have

[TABLE]

Now we take $M\to r$ and $N\to t$ in Eq. (3.1) and rewrite it as

[TABLE]

We then multiply both sides by $q^{r}/(1-q^{r})$ and sum over $r$

[TABLE]

From the foregoing argument, we therefore have

[TABLE]

Hence

[TABLE]

∎

Acknowledgements

I would like to thank George E. Andrews and Ae Ja Yee for some helpful discussions.