- Email: [email protected]
Refinements of the results on partitions and overpartitions with bounded part differences
Discrete Mathematics xxx (xxxx) xxx
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Refinements of the results on partitions and overpartitions with bounded part differences Bernard L.S. Lin School of Science, Jimei University, Xiamen 361021, P.R. China
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info
a b s t r a c t Recently, partitions with fixed or bounded differences between largest and smallest parts have attracted a lot of attention. In this paper, we first give a simple combinatorial proof of Breuer and Kronholm’s identity. Inspired by it, we construct a useful bijection to produce refinements of the results for partitions and overpartitions with bounded differences between largest and smallest parts. Consequently, we obtain Chern’s curious identity in a combinatorial manner. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 18 July 2019 Received in revised form 11 August 2019 Accepted 21 September 2019 Available online xxxx Keywords: Partition Overpartition Difference between largest and smallest parts
1. Introduction A partition of a positive integer n is a weakly decreasing sequence of positive integers (λ1 , λ2 , . . . , λℓ ) such that n = λ1 + λ2 + · · · + λℓ . The weight of λ, denoted by |λ|, is the sum of all parts, and the length of λ, denoted by ℓ(λ), is the number of parts. Let p(n, t) be the number of partitions of n with fixed difference t between largest and smallest parts. For t > 1, Andrews, Beck and Robbins [1] proved the following result ∞ ∑
p(n, t)qn =
n=1
qt −1 (1 − q)
(1 −
qt
) 1−
(
qt −1
)−
qt −1 (1 − q)
(1 −
qt
) 1−
(
qt −1
)
(q; q)t
+(
1−
qt qt −1
)
(q; q)t
,
(1.1)
where, as usual, we adopt the standard q-series notation (a; q)n =
n ∏
(1 − aqk−1 ), n ∈ N.
k=1
Motivated by their work, Breuer and Kronholm [2] studied the number p˜ (n, t) of partitions of n with bounded difference t between largest and smallest parts. For t ≥ 1, they obtained the following generating function
∑ n≥1
p˜ (n, t)q = n
1 1 − qt
(
1 (q; q)t
)
−1 .
(1.2)
Their approach is geometric and the main tool used is polyhedral cones. In a subsequent paper, Chapman [3] provided a simpler proof of (1.2), using only elementary q-series manipulation as deep as the q-binomial theorem. After that, overpartitions with bounded differences between largest and smallest parts are investigated. An overpartition of n is a partition of n where the first occurrence of a number may be overlined. For example, there are four E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111676 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
2
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
overpartitions of 2, namely, (2), (2), (1, 1), (1, 1). Let gt (n) be the number of overpartitions of n in which the difference between largest and smallest parts is at most t, and if the difference is exactly t, then the largest part cannot be overlined. For t ≥ 1, Chern [5] established the following generating function ∞ ∑
gt (n)q =
n=1
(
1
n
(−q; q)t
1 − qt
(q; q)t
)
−1 .
(1.3)
Define gt (m, n) to be the number of overpartitions counted by gt (n) with exactly m overlined parts. Then, Chern and Yee [6] presented the following refinement of (1.3) ∞ ∑ ∞ ∑
gt (m, n)z m qn =
n=1 m=0
(
1
(−zq; q)t
1 − qt
(q; q)t
) −1 .
(1.4)
By taking z → 0 and z → 1 respectively, (1.4) reduces to (1.2) and (1.3). More recently, Chern [4] found an interesting identity
∑ (1 − α qr )(1 − α qr +1 ) · · · (1 − α qr +t −2 ) (1 − β qr )(1 − β qr +1 ) · · · (1 − β qr +t )
r ≥1
q =
(
q
r
(β q − α )(1 − qt )
(α; q)t (β q; q)t
)
−1 ,
(1.5)
where t is a fixed positive integer, and α, β, q are complex variables with |q| < 1, q ̸ = 0, α ̸ = β q and (β q; q)t ̸ = 0. Employing this identity, one can prove (1.2) and the results in [5,6] easily. In addition, Chern [4] applied (1.5) to derive the generating function of the number po2t (n) of partitions of n into odd parts with bounded difference 2t between largest and smallest parts. To generalize this result, the present author [7] studied the number bk (n, t) of k-regular partitions where the difference between largest and smallest parts is at most kt and got the following neat formula ∞ ∑
bk (n, t)qn =
n=1
(
1 1 − qkt
(qk ; qk )t
) −1 .
(q; q)kt
(1.6)
The motivation of this paper is to present a simple combinatorial proof of (1.2). Inspired by this work, we obtain the following refinement of (1.4) and (1.6). Theorem 1.1. Given a positive integer t, let gt (i, j, n) count the number of overpartitions of n in which there are exactly i overlined parts and j nonoverlined parts, the difference between largest and smallest parts is at most t, and if the difference between largest and smallest parts is exactly t, then the largest part cannot be overlined. Then, we have ∞ ∑ ∞ ∑ ∞ ∑
gt (i, j, n)x y q =
n=1 i=0 j=0
(
1
i j n
(−xq; q)t
1 − qt
(yq; q)t
)
−1 .
(1.7)
Remark. Eq. (1.4) follows from (1.7) by taking y = 1. Theorem 1.2. For a fixed positive integer t, let bk (i, j, n, t) count the number of overpartitions of n in which no part is divisible by k and there are exactly i overlined parts and j nonoverlined parts, the difference between largest and smallest parts is at most kt, and if the difference between largest and smallest parts is exactly kt, then the largest part cannot be overlined. Then, we have ∞ ∑ ∞ ∑ ∞ ∑
bk (i, j, n, t)xi yj qn =
n=1 i=0 j=0
1 1 − qkt
(
(−xq; q)kt (yqk ; qk )t (−xqk ; qk )t (yq; q)kt
) −1 .
(1.8)
Remark. Setting x = 0 and y = 1, (1.8) degenerates to (1.6). The rest of the paper is organized as follows. In Section 2, we first give a simple proof of (1.2) applying combinatorial analysis, and then construct a bijection which yields a refinement of (1.2). In Section 3, we aim to present combinatorial proof of Theorems 1.1 and 1.2. As a consequence, we obtain Chern’s identity (1.5) in a combinatorial manner. 2. Refinement of Breuer and Kronholm’s identity In this section, we first give a simple proof of (1.2) using combinatorial analysis, and then present a refinement of (1.2) accompanied by a combinatorial proof. Proof of (1.2). Let r be the smallest part, then the largest part is at most r + t. Thus, ∞ ∑ n=1
p˜ (n, t)qn =
∞ ∑ r =1
qr
1
1 − qr 1 − qr +1
···
1 1 − qr +t
.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
3
On the other hand, we consider such partitions from the largest part. If the largest part is at most t, then there is no restriction on the smallest part. If the largest part is greater than t, assuming being t + s and s ≥ 1, then the smallest part must be at least s. Recall that the generating function for nonempty partitions whose largest part is at most t satisfies 1 (q; q)t
− 1.
Based on the above analysis, we see that ∞ ∑
p˜ (n, t)qn =
n=1
=
=
1
−1+
(q; q)t
∞ ∑ s=1
1
− 1 + qt
(q; q)t
qt +s
∞ ∑
qs
− 1 + qt
(q; q)t
∞ ∑
···
1
1 − qs+t 1 − qs+t −1
s=1
1
1
1 − qt +s 1 − qt +s−1
1 1 − qs
···
1 1 − qs
p˜ (n, t)qn ,
n=1
which yields the desired (1.2).
■
From the above combinatorial analysis, we can construct a simple bijection, which implies a refinement of (1.2) and can be used to prove Theorems 1.1 and 1.2 in Section 3. Denote by Pt the set of all nonempty partitions with the difference between largest part and smallest parts at most t. Let Pt ,≤ and Pt ,> be the subsets of Pt with the largest part being at most t and greater than t, respectively. We now define a bijection from Pt ,> to Pt . Given a partition λ = (λ1 , λ2 , . . . , λr ) ∈ Pt ,> , define ϕ (λ) = (λ2 , λ3 , . . . , λr , λ1 − t). We claim that ϕ (λ) ∈ Pt . Due to λ ∈ Pt ,> , we have λ1 > t and λ1 − λr ≤ t, which means λr ≥ λ1 − t > 0 and λ2 − (λ1 − t) ≤ t. Thus, ϕ (λ) ∈ Pt , ℓ(λ) = ℓ(ϕ (λ)) and |λ| = t + |ϕ (λ)|. Clearly, the above process is invertible. We are now in a position to prove the following refinement. Theorem 2.1. Let p˜ (m, n, t) be the number of partitions of n into m parts with bounded difference t between largest and smallest parts. Then, ∞ ∑ ∞ ∑
p˜ (m, n, t)z m qn =
n=1 m=1
(
1 1 − qt
1 (zq; q)t
) −1 .
(2.1)
Proof. It follows from the bijection ϕ that
∑
z ℓ(λ) q|λ| =
λ∈Pt ,>
z ℓ(ϕ (λ)) qt +|ϕ (λ)| = qt
∑
∑
z ℓ(λ) q|λ| .
λ∈Pt
ϕ (λ)∈Pt
Since Pt is the disjoint union of Pt ,≤ and Pt ,> , we have
∑
z ℓ(λ) q|λ| =
λ∈Pt
z ℓ(λ) q|λ| +
∑ λ∈Pt ,≤
∑
z ℓ(λ) q|λ| .
λ∈Pt ,>
Combining the above two identities, we obtain
∑
z ℓ(λ) q|λ| =
λ∈Pt
1 1−
z ℓ(λ) q|λ| .
∑ qt
λ∈Pt ,≤
It can be computed that
∑
z ℓ(λ) q|λ| =
λ∈Pt ,≤
1
− 1.
(zq; q)t
Therefore,
∑
z
ℓ(λ) |λ|
λ∈Pt
q
=
1 1 − qt
(
)
1 (zq; q)t
−1 .
Note that ∞ ∑ ∞ ∑
p˜ (m, n, t)z m qn =
n=1 m=1
which completes the proof.
∑
z ℓ(λ) q|λ| ,
λ∈Pt
■
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
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B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
3. Combinatorial proof of Theorems 1.1 and 1.2 Since the proof of Theorem 1.2 is similar to that of Theorem 1.1, we only focus on the proof of Theorem 1.1 here. Denote by P t the set of all nonempty overpartitions where the difference between largest and smallest parts is at most t, and if the difference between largest and smallest parts is exactly t, then the largest part cannot be overlined. Similarly, let P t ,≤ and P t ,> be the subsets of P t with the largest part being at most t and greater than t, respectively. Given an overpartition π = (π1 , π2 , . . . , πr ) ∈ P t ,> , define ψ (π ) = (π2 , π3 , . . . , πr , π1 − t), where the part π1 − t is overlined if and only if π1 is overlined. If the largest part π1 and the smallest part πr differ by t, i.e., πr = π1 − t, then π1 cannot be overlined, and neither can π1 − t, so ψ (π ) = (π2 , . . . , πr , π1 − t) is an overpartition in P t . It is not hard to see that the map ψ induces a bijection between P t ,> and P t , which preserves the number of overlined parts and nonoverlined parts, and decreases the weight by t. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. For an overpartition π , let o(π ) and uo(π ) be the number of overlined and nonoverlined parts in π , respectively. With the help of the bijection ψ , we see that
∑
xo(π ) yuo(π ) q|π | =
π ∈P t
∑
xo(π ) yuo(π ) q|π | +
π ∈P t ,≤
∑
=
xo(π ) yuo(π ) q|π |
∑ π ∈P t ,>
xo(π ) yuo(π ) q|π | + qt
π ∈P t ,≤
∑
xo(π ) yuo(π ) q|π| .
π ∈P t
It is straightforward to see that xo(π ) yuo(π ) q|π| =
∑ π ∈P t ,≤
(−xq; q)t (yq; q)t
− 1.
Combining the above two equations produces
∑
xo(π ) yuo(π ) q|π | =
π ∈P t
1 1 − qt
(
(−xq; q)t (yq; q)t
) −1 ,
which can be restated as ∞ ∑ ∞ ∑ ∞ ∑
gt (i, j, n)x y q = i j n
n=1 i=0 j=0
This completes the proof.
1
(
(−xq; q)t
1 − qt
(yq; q)t
)
−1 .
■
As a consequence, we can prove Chern’s identity (1.5) in a natural combinatorial way. Corollary 3.1. ∞ ∑ (1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) r =1
(1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
qr =
(
1 (1 − qt )(x + y)
(−xq; q)t (yq; q)t
) −1 .
(3.1)
Proof. For r ≥ 1, let P r ,t be the subset of P t consisting of those overpartitions with the smallest part being r. It is clear that Pt =
⋃
P r ,t
r ≥1
is a disjoint set-theoretic partitioning of P t . For π = (π1 , . . . , πℓ ) ∈ P r ,t , it follows from the definition that π1 ≤ r + t and π1 cannot be overlined if π1 = r + t. The first appearance of r can be overlined or not. An exact combinatorial analysis shows that
∑
xo(π ) yuo(π ) q|π | =
π ∈P r ,t
(1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) (1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
(xqr + yqr ),
which implies ∞ ∑ ∞ ∑ ∞ ∑ n=1 i=0 j=0
gt (i, j, n)xi yj qn =
∞ ∑ ∑
xo(π ) yuo(π ) q|π|
r =1 π ∈P r ,t
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
=
∞ ∑ (1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) r =1
(1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
The desired result follows from Theorem 1.1 immediately.
5
(xqr + yqr ).
■
Remark. Replacing x by −α/q and y by β in (3.1) yields Chern’s identity (1.5). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The author would like to thank the referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11871246), the Natural Science Foundation of Fujian Province of China (No. 2019J01328), and the Program for New Century Excellent Talents in Fujian Province University (No. B17160). References [1] G.E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, Proc. Amer. Math. Soc. 143 (2015) 4283–4289. [2] F. Breuer, B. Kronholm, A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of andrews, beck, and robbins, Res. Number Theory 2 (2016) Article 2. [3] R. Chapman, Partitions with bounded differences between largest and smallest parts, Australas. J. Combin. 64 (2016) 376–378. [4] S. Chern, A curious identity and its applications to partitions with bounded part differences, New Zealand J. Math. 47 (2017) 23–26. [5] S. Chern, An overpartition analogue of partitions with bounded differences between largest and smallest parts, Discrete Math. 340 (2017) 2834–2839. [6] S. Chern, A.J. Yee, Overpartitions with bounded part differences, European J. Combin. 70 (2018) 317–324. [7] B.L.S. Lin, k-regular partitions with bounded differences between largest and smallest parts, submitted for publication.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Refinements of the results on partitions and overpartitions with bounded part differences Bernard L.S. Lin School of Science, Jimei University, Xiamen 361021, P.R. China
article
info
a b s t r a c t Recently, partitions with fixed or bounded differences between largest and smallest parts have attracted a lot of attention. In this paper, we first give a simple combinatorial proof of Breuer and Kronholm’s identity. Inspired by it, we construct a useful bijection to produce refinements of the results for partitions and overpartitions with bounded differences between largest and smallest parts. Consequently, we obtain Chern’s curious identity in a combinatorial manner. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 18 July 2019 Received in revised form 11 August 2019 Accepted 21 September 2019 Available online xxxx Keywords: Partition Overpartition Difference between largest and smallest parts
1. Introduction A partition of a positive integer n is a weakly decreasing sequence of positive integers (λ1 , λ2 , . . . , λℓ ) such that n = λ1 + λ2 + · · · + λℓ . The weight of λ, denoted by |λ|, is the sum of all parts, and the length of λ, denoted by ℓ(λ), is the number of parts. Let p(n, t) be the number of partitions of n with fixed difference t between largest and smallest parts. For t > 1, Andrews, Beck and Robbins [1] proved the following result ∞ ∑
p(n, t)qn =
n=1
qt −1 (1 − q)
(1 −
qt
) 1−
(
qt −1
)−
qt −1 (1 − q)
(1 −
qt
) 1−
(
qt −1
)
(q; q)t
+(
1−
qt qt −1
)
(q; q)t
,
(1.1)
where, as usual, we adopt the standard q-series notation (a; q)n =
n ∏
(1 − aqk−1 ), n ∈ N.
k=1
Motivated by their work, Breuer and Kronholm [2] studied the number p˜ (n, t) of partitions of n with bounded difference t between largest and smallest parts. For t ≥ 1, they obtained the following generating function
∑ n≥1
p˜ (n, t)q = n
1 1 − qt
(
1 (q; q)t
)
−1 .
(1.2)
Their approach is geometric and the main tool used is polyhedral cones. In a subsequent paper, Chapman [3] provided a simpler proof of (1.2), using only elementary q-series manipulation as deep as the q-binomial theorem. After that, overpartitions with bounded differences between largest and smallest parts are investigated. An overpartition of n is a partition of n where the first occurrence of a number may be overlined. For example, there are four E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111676 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
2
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
overpartitions of 2, namely, (2), (2), (1, 1), (1, 1). Let gt (n) be the number of overpartitions of n in which the difference between largest and smallest parts is at most t, and if the difference is exactly t, then the largest part cannot be overlined. For t ≥ 1, Chern [5] established the following generating function ∞ ∑
gt (n)q =
n=1
(
1
n
(−q; q)t
1 − qt
(q; q)t
)
−1 .
(1.3)
Define gt (m, n) to be the number of overpartitions counted by gt (n) with exactly m overlined parts. Then, Chern and Yee [6] presented the following refinement of (1.3) ∞ ∑ ∞ ∑
gt (m, n)z m qn =
n=1 m=0
(
1
(−zq; q)t
1 − qt
(q; q)t
) −1 .
(1.4)
By taking z → 0 and z → 1 respectively, (1.4) reduces to (1.2) and (1.3). More recently, Chern [4] found an interesting identity
∑ (1 − α qr )(1 − α qr +1 ) · · · (1 − α qr +t −2 ) (1 − β qr )(1 − β qr +1 ) · · · (1 − β qr +t )
r ≥1
q =
(
q
r
(β q − α )(1 − qt )
(α; q)t (β q; q)t
)
−1 ,
(1.5)
where t is a fixed positive integer, and α, β, q are complex variables with |q| < 1, q ̸ = 0, α ̸ = β q and (β q; q)t ̸ = 0. Employing this identity, one can prove (1.2) and the results in [5,6] easily. In addition, Chern [4] applied (1.5) to derive the generating function of the number po2t (n) of partitions of n into odd parts with bounded difference 2t between largest and smallest parts. To generalize this result, the present author [7] studied the number bk (n, t) of k-regular partitions where the difference between largest and smallest parts is at most kt and got the following neat formula ∞ ∑
bk (n, t)qn =
n=1
(
1 1 − qkt
(qk ; qk )t
) −1 .
(q; q)kt
(1.6)
The motivation of this paper is to present a simple combinatorial proof of (1.2). Inspired by this work, we obtain the following refinement of (1.4) and (1.6). Theorem 1.1. Given a positive integer t, let gt (i, j, n) count the number of overpartitions of n in which there are exactly i overlined parts and j nonoverlined parts, the difference between largest and smallest parts is at most t, and if the difference between largest and smallest parts is exactly t, then the largest part cannot be overlined. Then, we have ∞ ∑ ∞ ∑ ∞ ∑
gt (i, j, n)x y q =
n=1 i=0 j=0
(
1
i j n
(−xq; q)t
1 − qt
(yq; q)t
)
−1 .
(1.7)
Remark. Eq. (1.4) follows from (1.7) by taking y = 1. Theorem 1.2. For a fixed positive integer t, let bk (i, j, n, t) count the number of overpartitions of n in which no part is divisible by k and there are exactly i overlined parts and j nonoverlined parts, the difference between largest and smallest parts is at most kt, and if the difference between largest and smallest parts is exactly kt, then the largest part cannot be overlined. Then, we have ∞ ∑ ∞ ∑ ∞ ∑
bk (i, j, n, t)xi yj qn =
n=1 i=0 j=0
1 1 − qkt
(
(−xq; q)kt (yqk ; qk )t (−xqk ; qk )t (yq; q)kt
) −1 .
(1.8)
Remark. Setting x = 0 and y = 1, (1.8) degenerates to (1.6). The rest of the paper is organized as follows. In Section 2, we first give a simple proof of (1.2) applying combinatorial analysis, and then construct a bijection which yields a refinement of (1.2). In Section 3, we aim to present combinatorial proof of Theorems 1.1 and 1.2. As a consequence, we obtain Chern’s identity (1.5) in a combinatorial manner. 2. Refinement of Breuer and Kronholm’s identity In this section, we first give a simple proof of (1.2) using combinatorial analysis, and then present a refinement of (1.2) accompanied by a combinatorial proof. Proof of (1.2). Let r be the smallest part, then the largest part is at most r + t. Thus, ∞ ∑ n=1
p˜ (n, t)qn =
∞ ∑ r =1
qr
1
1 − qr 1 − qr +1
···
1 1 − qr +t
.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
3
On the other hand, we consider such partitions from the largest part. If the largest part is at most t, then there is no restriction on the smallest part. If the largest part is greater than t, assuming being t + s and s ≥ 1, then the smallest part must be at least s. Recall that the generating function for nonempty partitions whose largest part is at most t satisfies 1 (q; q)t
− 1.
Based on the above analysis, we see that ∞ ∑
p˜ (n, t)qn =
n=1
=
=
1
−1+
(q; q)t
∞ ∑ s=1
1
− 1 + qt
(q; q)t
qt +s
∞ ∑
qs
− 1 + qt
(q; q)t
∞ ∑
···
1
1 − qs+t 1 − qs+t −1
s=1
1
1
1 − qt +s 1 − qt +s−1
1 1 − qs
···
1 1 − qs
p˜ (n, t)qn ,
n=1
which yields the desired (1.2).
■
From the above combinatorial analysis, we can construct a simple bijection, which implies a refinement of (1.2) and can be used to prove Theorems 1.1 and 1.2 in Section 3. Denote by Pt the set of all nonempty partitions with the difference between largest part and smallest parts at most t. Let Pt ,≤ and Pt ,> be the subsets of Pt with the largest part being at most t and greater than t, respectively. We now define a bijection from Pt ,> to Pt . Given a partition λ = (λ1 , λ2 , . . . , λr ) ∈ Pt ,> , define ϕ (λ) = (λ2 , λ3 , . . . , λr , λ1 − t). We claim that ϕ (λ) ∈ Pt . Due to λ ∈ Pt ,> , we have λ1 > t and λ1 − λr ≤ t, which means λr ≥ λ1 − t > 0 and λ2 − (λ1 − t) ≤ t. Thus, ϕ (λ) ∈ Pt , ℓ(λ) = ℓ(ϕ (λ)) and |λ| = t + |ϕ (λ)|. Clearly, the above process is invertible. We are now in a position to prove the following refinement. Theorem 2.1. Let p˜ (m, n, t) be the number of partitions of n into m parts with bounded difference t between largest and smallest parts. Then, ∞ ∑ ∞ ∑
p˜ (m, n, t)z m qn =
n=1 m=1
(
1 1 − qt
1 (zq; q)t
) −1 .
(2.1)
Proof. It follows from the bijection ϕ that
∑
z ℓ(λ) q|λ| =
λ∈Pt ,>
z ℓ(ϕ (λ)) qt +|ϕ (λ)| = qt
∑
∑
z ℓ(λ) q|λ| .
λ∈Pt
ϕ (λ)∈Pt
Since Pt is the disjoint union of Pt ,≤ and Pt ,> , we have
∑
z ℓ(λ) q|λ| =
λ∈Pt
z ℓ(λ) q|λ| +
∑ λ∈Pt ,≤
∑
z ℓ(λ) q|λ| .
λ∈Pt ,>
Combining the above two identities, we obtain
∑
z ℓ(λ) q|λ| =
λ∈Pt
1 1−
z ℓ(λ) q|λ| .
∑ qt
λ∈Pt ,≤
It can be computed that
∑
z ℓ(λ) q|λ| =
λ∈Pt ,≤
1
− 1.
(zq; q)t
Therefore,
∑
z
ℓ(λ) |λ|
λ∈Pt
q
=
1 1 − qt
(
)
1 (zq; q)t
−1 .
Note that ∞ ∑ ∞ ∑
p˜ (m, n, t)z m qn =
n=1 m=1
which completes the proof.
∑
z ℓ(λ) q|λ| ,
λ∈Pt
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Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
4
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
3. Combinatorial proof of Theorems 1.1 and 1.2 Since the proof of Theorem 1.2 is similar to that of Theorem 1.1, we only focus on the proof of Theorem 1.1 here. Denote by P t the set of all nonempty overpartitions where the difference between largest and smallest parts is at most t, and if the difference between largest and smallest parts is exactly t, then the largest part cannot be overlined. Similarly, let P t ,≤ and P t ,> be the subsets of P t with the largest part being at most t and greater than t, respectively. Given an overpartition π = (π1 , π2 , . . . , πr ) ∈ P t ,> , define ψ (π ) = (π2 , π3 , . . . , πr , π1 − t), where the part π1 − t is overlined if and only if π1 is overlined. If the largest part π1 and the smallest part πr differ by t, i.e., πr = π1 − t, then π1 cannot be overlined, and neither can π1 − t, so ψ (π ) = (π2 , . . . , πr , π1 − t) is an overpartition in P t . It is not hard to see that the map ψ induces a bijection between P t ,> and P t , which preserves the number of overlined parts and nonoverlined parts, and decreases the weight by t. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. For an overpartition π , let o(π ) and uo(π ) be the number of overlined and nonoverlined parts in π , respectively. With the help of the bijection ψ , we see that
∑
xo(π ) yuo(π ) q|π | =
π ∈P t
∑
xo(π ) yuo(π ) q|π | +
π ∈P t ,≤
∑
=
xo(π ) yuo(π ) q|π |
∑ π ∈P t ,>
xo(π ) yuo(π ) q|π | + qt
π ∈P t ,≤
∑
xo(π ) yuo(π ) q|π| .
π ∈P t
It is straightforward to see that xo(π ) yuo(π ) q|π| =
∑ π ∈P t ,≤
(−xq; q)t (yq; q)t
− 1.
Combining the above two equations produces
∑
xo(π ) yuo(π ) q|π | =
π ∈P t
1 1 − qt
(
(−xq; q)t (yq; q)t
) −1 ,
which can be restated as ∞ ∑ ∞ ∑ ∞ ∑
gt (i, j, n)x y q = i j n
n=1 i=0 j=0
This completes the proof.
1
(
(−xq; q)t
1 − qt
(yq; q)t
)
−1 .
■
As a consequence, we can prove Chern’s identity (1.5) in a natural combinatorial way. Corollary 3.1. ∞ ∑ (1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) r =1
(1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
qr =
(
1 (1 − qt )(x + y)
(−xq; q)t (yq; q)t
) −1 .
(3.1)
Proof. For r ≥ 1, let P r ,t be the subset of P t consisting of those overpartitions with the smallest part being r. It is clear that Pt =
⋃
P r ,t
r ≥1
is a disjoint set-theoretic partitioning of P t . For π = (π1 , . . . , πℓ ) ∈ P r ,t , it follows from the definition that π1 ≤ r + t and π1 cannot be overlined if π1 = r + t. The first appearance of r can be overlined or not. An exact combinatorial analysis shows that
∑
xo(π ) yuo(π ) q|π | =
π ∈P r ,t
(1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) (1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
(xqr + yqr ),
which implies ∞ ∑ ∞ ∑ ∞ ∑ n=1 i=0 j=0
gt (i, j, n)xi yj qn =
∞ ∑ ∑
xo(π ) yuo(π ) q|π|
r =1 π ∈P r ,t
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.
B.L.S. Lin / Discrete Mathematics xxx (xxxx) xxx
=
∞ ∑ (1 + xqr +1 )(1 + xqr +2 ) · · · (1 + xqr +t −1 ) r =1
(1 − yqr )(1 − yqr +1 ) · · · (1 − yqr +t )
The desired result follows from Theorem 1.1 immediately.
5
(xqr + yqr ).
■
Remark. Replacing x by −α/q and y by β in (3.1) yields Chern’s identity (1.5). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The author would like to thank the referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11871246), the Natural Science Foundation of Fujian Province of China (No. 2019J01328), and the Program for New Century Excellent Talents in Fujian Province University (No. B17160). References [1] G.E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, Proc. Amer. Math. Soc. 143 (2015) 4283–4289. [2] F. Breuer, B. Kronholm, A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of andrews, beck, and robbins, Res. Number Theory 2 (2016) Article 2. [3] R. Chapman, Partitions with bounded differences between largest and smallest parts, Australas. J. Combin. 64 (2016) 376–378. [4] S. Chern, A curious identity and its applications to partitions with bounded part differences, New Zealand J. Math. 47 (2017) 23–26. [5] S. Chern, An overpartition analogue of partitions with bounded differences between largest and smallest parts, Discrete Math. 340 (2017) 2834–2839. [6] S. Chern, A.J. Yee, Overpartitions with bounded part differences, European J. Combin. 70 (2018) 317–324. [7] B.L.S. Lin, k-regular partitions with bounded differences between largest and smallest parts, submitted for publication.
Please cite this article as: B.L.S. Lin, Refinements of the results on partitions and overpartitions with bounded part differences, Discrete Mathematics (2019) 111676, https://doi.org/10.1016/j.disc.2019.111676.