Enumerating Parts of \(n\)-Color Partitions
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#A94 INTEGERS 25 (2025) ENUMERATING PARTS OF n-COLOR PARTITIONS Shruti Sharma Yadavindra Department of Sciences, Punjabi University Guru Kashi Campus, Talwandi Sabo, Punjab, India sharma [email protected] Amandeep Kaur Department of Mathematics, Punjabi University, Patiala, Punjab, India [email protected] Received: 2/3/25, Accepted: 9/25/25, Published: 11/5/25 Abstract In recent years, a number of authors have obtained numerous relations between the number of parts in restricted as well as unrestricted partitions and various functions from multiplicative number theory such as ϕ(m), τ(m), σ(m), etc. In this paper, we seek similar kinds of relations for the number of parts in various restricted and unrestricted n-color partitions. Here, in particular, these relations occur with divisor functions. 1. Introduction Apartition of a positive integer mis a sequence of positive integers whose sum is m. While the order of positive integers in a partition does not matter, we conventionally list them in non-increasing order for consistency. These positive integers are also called the summands or parts of a partition. Many important partition identities involve restrictions on the parts of a partition. A very old and famous identity due to Euler equates the number of partitions into odd parts to the number of partitions into distinct parts. Partition identities also arise from counting the number of parts in a partition. In the literature, the first such identity is credited to Stanley, and its generalization to Elder. To know more about the history of these identities, the reader is referred to [5]. Theorem 1 (Stanley’s theorem).The number of 1’s in the partitions of nequals the number of parts that appear at least once in a given partition of n, summed over all partitions of n. DOI: 10.5281/zenodo.17535215
INTEGERS: 25 (2025) 2 Theorem 2 (Elder’s theorem).The number of appearances of a part kin the partitions of nis equal to the number of parts that appear at least ktimes in a given partition of n, summed over all partitions of n. Andrews and Merca’s work [3] provides a further generalization of Elder’s theorem. In [4], the authors count the number of even parts in partitions into distinct parts. Merca [6] investigated specific restricted partitions, enumerating their parts and establishing relationships between these counts and the divisors of m. In [7,8], the authors obtained numerous relations between the number of parts in partitions (restricted as well as unrestricted) and various functions from multiplicative number theory such as ϕ(m), τ(m), σ(m), etc. In order to obtain these relations, they used the Lambert series, which is given by ∞ X m=1 amqm 1−qm,|q|<1, where the amfor m= 1,2,3, . . . are real or complex numbers. This series is a natural generalization of the formula ∞ X m=1 amqm 1−qm= ∞ X m=1 b±(m)qm, where b±(m) = X d|m (∓1)1+m/dad. When working with n-color partitions, it is natural for us to seek similar kinds of relations for the number of parts in various restricted and unrestricted n-color partitions. In this manuscript, however, we will focus only on the divisor functions. For a reader who is not exposed to the theory of n-color partitions, we include the definition of n-color partitions as given in [2]. An n-color partition of a positive integer is a partition, in which a part of size mcan occur in mdifferent colors denoted by subscripts m1, m2, . . . , mm. For example, the n-color partitions of 3 are 31,32,33,22+ 11,21+ 11,11+ 11+ 11. Let P(m) denote the number of partitions of m. Then, ∞ X m=0 P(m)qm=1 ∞ Q m=1 (1 −qm) . As a convention, we take P(0) as 1. Using similar notation, we write ∞ X m=0 C(m)qm=1 ∞ Q m=1 (1 −qm)m ,(1)
INTEGERS: 25 (2025) 3 where C(m) is the number of n-color partitions of mand C(0) = 1. As ordinary partitions have been studied with various restrictions, n-color partitions too can be considered with restrictions on the parts and colors. Generating functions for the number of n-color partitions with various restrictions were provided by Agarwal [1]. However, we take into account some of these n-color partitions with restrictions while studying the number of parts. We include the generating functions for C(D, m), C(O, m), and C(E, m), which denote, respectively, the number of n-color partitions into distinct parts, into odd parts, and into even parts of a positive integer m. Also, we take C(D, 0) = C(O, 0) = C(E, 0) = 1 as a convention. The generating functions for C(D, m), C(O, m), and C(E, m) are given by ∞ X m=0 C(D, m)qm= ∞ Y m=1 (1 + qm)m,(2) ∞ X m=0 C(O, m)qm= ∞ Y m=1 1 (1 −q2m−1)2m−1, ∞ X m=0 C(E, m)qm= ∞ Y m=1 1 (1 −q2m)2m. Let Ce−o(m) denote the number of n-color partitions of a positive integer minto an even number of parts minus the number of n-color partitions of a positive integer m into an odd number of parts. We can easily obtain the following generating function for Ce−o(m), where we take Ce−o(0) = 1: ∞ X m=0 Ce−o(m)qm= ∞ Y m=1 1 (1 + qm)m.(3) Now, we let Cod(m) denote the number of n-color partitions of mwherein odd parts are distinct and even parts are unrestricted and Ced(m) denote the number of n-color partitions of mwherein even parts are distinct and odd parts are unrestricted. Again, we take Cod(0) = 1 and Ced(0) = 1 to obtain the following generating functions for Cod(m) and Ced(m): ∞ X m=0 Cod(m)qm= ∞ Y m=1 (1 + q2m−1)2m−1 (1 −q2m)2m, ∞ X m=0 Ced(m)qm= ∞ Y m=1 (1 + q2m)2m (1 −q2m−1)2m−1. In Section 2, we enumerate the parts of n-color partitions without restrictions and investigate their relationship with various divisor functions. In Section 3, we investigate similar relations while counting the number of parts of n-color partitions with restrictions.
INTEGERS: 25 (2025) 4 2. Number of Parts in n-Color Partitions and Divisor Functions We start this section by summing the number of parts for all the n-color partitions into an odd (even) number of parts which provides us our first result. Theorem 3. Let To(m) (respectively Te(m)) denote the sum of the number of parts, where the sum is taken over all the n-color partitions of a positive integer minto an odd (respectively even)number of parts. Then for |q|<1, ∞ X m=1 mqm 1∓qm= ∞ Y m=1 (1∓qm)m ∞ X m=1 (To(m)±Te(m))qm. Proof. Considering the generating function for n-color partitions and introducing a new variable xwhich keeps track of the number of parts, we have ∞ X m=1 (To(m)±Te(m))qm=d dxx=±1(1 −xq)−1(1 −xq2)−2· · · =1 ∞ Q m=1 (1∓qm)m ∞ X m=1 mqm 1∓qm. This concludes our proof. We also have the following results from sequences A000593, A146076, and A000203 in the OEIS [9]: ∞ X m=1 mqm 1 + qm= ∞ X m=1 (2m−1)q2m−1 1−q2m−1= ∞ X m=1 σodd(m)qm,(4) ∞ X m=1 2mq2m 1−q2m= ∞ X m=1 σeven(m)qm,(5) where σeven(m) (respectively, σodd(m)) is the sum of even divisors (respectively, odd divisors) of m. Also, ∞ X m=1 (2m−1)q2m−1 1 + q2m−1= ∞ X m=1 (−1)m+1σodd(m)qm and ∞ X m=1 mqm 1−qm= ∞ X m=1 σ(m)qm,|q|<1,(6) where σ(m) denotes the sum of divisors of m. Now, using Equations (1), (2), (4), and (6), we get Corollary 1and Corollary 2to Theorem 3with the help of convolution properties.
INTEGERS: 25 (2025) 5 Corollary 1. For any positive integer m, To(m) + Te(m) = m X k=1 σ(k)C(m−k).(7) We take an example to verify Corollary 1for m= 4. Example 1. n-color partitions of 4 into an odd number of parts are 41,42,43,44,22+ 11+ 11,21+ 11+ 11, and into an even number of parts are 31+ 11,32+ 11,33+ 11,22+ 22,22+ 21,2+1+11,11+ 11+ 11+ 11. Thus, To(4) = 10 and Te(4) = 16. We know σ(1) = 1, σ(2) = 3, σ(3) = 4, and σ(4) = 7. Also, C(3) = 6, C(2) = 3, C(1) = 1, and C(0) = 1. On substituting all these values and m= 4 in (7), we see that To(4) + Te(4) = 4 X k=1 σ(k)C(4 −k). Corollary 2. For any positive integer m, σodd(m) = m X k=1 (To(k)−Te(k)) C(D, m −k). In Theorem 4, we count the number of even parts in all the n-color partitions. Theorem 4. Let Ae(m)denote the sum of the number of even parts, where the sum is taken over all the n-color partitions of a positive integer m. Then for |q|<1, ∞ X m=1 2mq2m 1−q2m= ∞ Y m=1 (1 −qm)m ∞ X m=1 Ae(m)qm. Proof. Proceeding in a similar manner as in the previous theorem, we see that ∞ X m=1 Ae(m)qm= ∞ Y m=1 1 (1 −q2m−1)2m−1 d dxx=1 ∞ Y m=1 1 (1 −xq2m)2m = ∞ Y m=1 1 (1 −qm)m ∞ X m=1 2mq2m 1−q2m. This completes the proof. As a direct consequence of Equations (1) and (5), we obtain the following corollary of Theorem 4.
INTEGERS: 25 (2025) 6 Corollary 3. For any positive integer m, Ae(m) = m X k=1 σeven(k)C(m−k). Theorem 5is similar to Theorem 4with the only difference being that we now count the number of odd parts. Theorem 5. Let Ao(m)represent the sum of the number of odd parts, where the sum is taken over all the n-color partitions of a positive integer m. Then for |q|<1, ∞ X m=1 (2m−1)q2m−1 1−q2m−1= ∞ Y m=1 (1 −qm)m ∞ X m=1 Ao(m)qm. Corollary 4. For any positive integer m, Ao(m) = m X k=1 σodd(k)C(m−k). Our next result is about counting the number of parts with multiple occurrences. Theorem 6. For positive integers mand t, let Nt(m)denote the sum of the number of distinct parts with multiplicity at least t, where the sum is taken over all the ncolor partitions of m. Then, ∞ X m=1 mqmt = ∞ Y m=1 (1 −qm)m ∞ X m=1 Nt(m)qm, m ≥t. Proof. We can obtain the generating function for Nt(m) by introducing a variable x to account for the number of parts with multiplicity at least t, then taking derivative with respect to xand finally substituting x= 1. Hence, ∞ X m=1 Nt(m)qm=d dxx=1 ∞ Y m=1 1 + qm+· · · +q(t−1)m+xqmt 1−qmm = ∞ Y m=1 1 (1 −qm)m ∞ X m=1 mqmt. Corollary 5. For positive integers mand t, Nt(m) = ⌊m/t⌋ X k=1 kC(m−tk), m ≥t.
INTEGERS: 25 (2025) 7 In Theorem 7, we count number of parts congruent to k(mod r) for some fixed positive integers kand r. Theorem 7. For r≥1and 0< k < r, let C(kmod r, m)denote the sum of the number of parts congruent to k (mod r), where the sum is taken over all the n-color partitions of a positive integer m. Then, ∞ X m=0 (rm +k)qrm+k 1−qrm+k= ∞ Y m=1 (1 −qm)m ∞ X m=1 C(kmod r, m)qm. Proof. In the following two variable generating function, xkeeps track of parts congruent to k(mod r) in all the n-color partitions: ∞ Y i=1 i≡k(mod r) 1 (1 −qi)i ∞ Y j=1 j≡k(mod r) 1 (1 −xqj)j. By applying d dxx=1 to the above expression, we obtain ∞ X m=1 C(k(mod r), m)qm= ∞ Y i=1 i≡k(mod r) 1 (1 −qi)i ∞ Y i=1 i≡k(mod r) 1 (1 −qi)i ∞ X m=0 (rm +k)qrm+k 1−qrm+k = ∞ Y m=1 1 (1 −qm)m ∞ X m=0 (rm +k)qrm+k 1−qrm+k. Corollary 6. For integers m > 0,r≥1and 0< k < r, C(kmod r, m) = m X j=1 X rd+k|j (rd +k) C(m−j). 3. Number of Parts in n-Color Partitions with Restrictions and Divisor Functions In this section, we focus on counting the number of parts in some restricted n-color partitions and we start by considering n-color partitions into distinct parts. Theorem 8. Let Co(D, m) (respectively Ce(D, m)) denote the sum of the number of parts, where the sum is taken over all the n-color partitions of a positive integer minto an odd (respectively even)number of distinct parts. Then for |q|<1, ∞ X m=1 mqm 1±qm=1 ∞ Q m=1 (1±qm)m ∞ X m=1 (Co(D, m)±Ce(D, m)) qm.
INTEGERS: 25 (2025) 8 Proof. Proceeding in a manner similar to the proof of Theorem 3, we have ∞ X m=1 (Co(D, m)±Ce(D, m)) qm=d dxx=±1(1 + xq)(1 + xq2)2(1 + xq3)3· · · = ∞ Y m=1 (1 + xqm)mq 1 + xq +2q2 1 + xq2+· · · x=±1 = ∞ Y m=1 (1±qm)m ∞ X m=1 mqm 1±qm, which gives us the desired result. To obtain Corollaries 7-9to Theorem 8, we use Equations (1), (2), (3), (4), (6), and the Cauchy product of two power series. Corollary 7. For any positive integer m, Co(D, m) + Ce(D, m) = m X k=1 σodd(k)C(D, m −k). Corollary 8. For any positive integer m, σ(m) = m X k=1 (Co(D, k)−Ce(D, k)) C(m−k). Corollary 9. For any positive integer m, σodd(m) = m X k=1 C(D, k)Ce−o(m−k). Proceeding in a similar manner as in Theorem 8, we can prove Theorem 9. Theorem 9. Let Q(O, m)represent sum of the number of parts, where the sum is taken over all the n-color partitions of a positive integer minto odd parts. Then for |q|<1, ∞ X m=1 (2m−1)q2m−1 1−q2m−1= ∞ Y m=1 (1 −q2m−1)2m−1 ∞ X m=1 Q(O, m)qm. Corollary 10. For any positive integer m, Q(O, m) = m X k=1 σodd(k)C(O, m −k).
INTEGERS: 25 (2025) 9 Similar results can be proved for the number of parts in all the n-color partitions of a positive integer into even parts. Our next theorem is related to the set of n-color partitions with distinct odd parts and unrestricted even parts. Theorem 10. Let Qod(m)represent the sum of the number of parts, where the sum is taken over all the n-color partitions of a positive integer mwith distinct odd parts and unrestricted even parts. Then for |q|<1, ∞ X m=1 (2m−1)q2m−1 1 + q2m−1= ∞ Y m=1 (1 −q2m)2m (1 + q2m−1)2m−1 ∞ X m=1 Qod(m)qm. Corollary 11. For a positive integer m, Qod(m) = m X k=1 X 2d−1|k (−1)1+k/2d−1(2d−1) Cod(m−k). Again, similar results can be proved for Qed(m), where Qed(m) represents the sum of the number of parts with the sum taken over all the n-color partitions of a non-negative integer into distinct even parts and unrestricted odd parts. Theorem 11 is also about counting the number of parts in n-color partitions into distinct parts but here we count some fixed part. Theorem 11. For j≥1, let Q(D, j, m)denote the sum of the number of j’s, where the sum is taken over all the n-color partitions of a positive integer minto distinct parts. Then for |q|<1, ∞ X m=1 Q(D, j, m)qm=jqj 1 + qj ∞ X m=0 C(D, m)qm. Proof. To prove this result, we introduce a variable xto count the total number of j’s in all the n-color partitions of minto distinct parts. Then, we take the derivative with respect to xand put x= 1. Hence, ∞ X m=1 Q(D, j, m)qm=d dxx=1 1 + xqjj∞ Y i=1 i=j (1 + qi)i =jqj 1 + qjY i≥1 (1 + qi)i =jqj 1 + qj ∞ X m=0 C(D, m)qm.