n -Color Partitions into Distinct Parts as Sums over Partitions

: The partitions in which the parts of size n can come in n different colors are known as n -color partitions. For r ∈ { 0,1 } , let QL r ( n ) be the number of n -color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for QL 0 ( n ) ± QL 1 ( n ) as sums over partitions of n in terms of binomial coefﬁcients. The ﬁrst kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the ﬁrst differences of QL 0 ( n ) ± QL 1 ( n ) and the partial sums of QL 0 ( n ) ± QL 1 ( n ) .


Introduction
Compositions and partitions are fascinating topics in number theory, and they have many applications in combinatorics and other fields.For example, compositions and partitions can be used to count the number of ways to arrange objects, encrypt messages, solve equations, and more.They are also related to other concepts such as Fibonacci numbers, modular arithmetic, and symmetric groups.Compositions and partitions are examples of how simple ideas can lead to rich and beautiful mathematics.
Compositions and partitions are two ways of writing an integer as a sum of positive integers, but they differ in how they treat the order of the terms.For example, 4 = 2 + 1 + 1 and 4 = 1 + 2 + 1 are two different compositions of 4, but they are the same partition of 4. In general, a composition of n is an ordered list of positive integers whose sum is n, and a partition of n is an unordered list of positive integers whose sum is n.
The concept of n-color partitions is a natural generalization of the concept of ordinary partitions.An n-color partition of a positive integer is a partition in which a part of size n can come in n different colors denoted by subscripts n 1 , n 2 , . . ., n n .The parts satisfy the following order: In this paper, we denote by QL(m) the number of n-color partitions of m into distinct parts.For convenience, we define QL(0) = 1.For example, there are sixteen n-color partitions into distinct parts of five: (5 5 ), (5 4 ), (5 3 ), (5 2 ), ( (4) We remark that n-color partitions were introduced to mathematics in 1987 by A. K. Agarwal and G. E. Andrews [2] for giving combinatorial interpretations of several q-series identities.We mention that n-color partitions were used indirectly in many studies of planar partitions before Andrews and Agarwal started studying n-color partitions [3][4][5].For further reading on n-color partitions, we refer to [6][7][8][9][10][11][12][13][14].
For r ∈ {0, 1}, we denote by QL r (m) the number of n-color partitions of m into distinct parts which have a number of parts congruent to r modulo 2. For example, there are ten n-color partitions of five into distinct parts with an even number of parts: and six n-color partitions of five into distinct parts with an odd number of parts: (5 5 ), (5 4 ), (5 3 ), (5 2 ), (5 1 ), (2 2 , 2 1 , 1 1 ).

It is clear that
Elementary techniques in the theory of partitions [1] give the following generating function: The expansions start as (1 + q n ) n = 1 + q + 2 q 2 + 5 q 3 + 8 q 4 + 16 q 5 + 28 q 6 + 49 q 7 + 83 and If in a partition of n we have t k parts of size k, then we can color these parts in distinct colors in ( k t k ) ways.This remark allows us to immediately derive the following formula.
Theorem 2. For n 0, The sum in the right-hand side of this equation runs over all the partitions of n, but not all terms are non-zero.Because for t k > k we have ( k t k ) = 0, in this sum we can consider only the partitions of n into at most k copies of parts of size k, for each k ∈ {1, 2, . . ., n}.On the other hand, it is known that the number of these partitions of n is equal to the number of partitions of n into non-pronic numbers (cannot be written as i(i + 1)) ( [15] A002378, A052335).This combinatorial interpretation follows easily if we take into account the following relation The left-hand side of this equation is the generating function for the number of partitions of n into at most k copies of parts of size k, while the right-hand side is the generating function for the number of partitions of n into parts which cannot be written as i(i + 1).Recall that the numbers that can be arranged to form a rectangle are called rectangular numbers (also known as pronic numbers).For example, the number 12 is a rectangular number because it is three rows by four columns.
There is a more general result which combines compositions and partitions, where our Theorem 2 is the first entry.Theorem 3. Let m be a positive integer.For n 0, We note that the sum on the left-hand side of this identity runs over all the compositions of n into exactly m parts, while the sum on the right-hand side runs over all the partitions of n into at most m • k copies of parts of size k, for each k ∈ {1, 2, . . ., n}.
Symmetry is an important concept in mathematics, and it plays an important role in the study of integer partitions.There are many interesting results and theorems related to integer partitions and symmetry (see, for example, [16]), and they have applications in many areas of mathematics and beyond.In this paper, we take into account specializations of the elementary symmetric functions in order to provide an analytic proof of Theorem 3. Our approach allows us to obtain other results involving n-color partitions into distinct parts.
In order to introduce the following result, we consider the sequence (a n ) n 1 , defined by Theorem 4. Let m be a positive integer.For n 0, In this context, we remark that the first differences of QL(n) can be expressed as a sum over the partitions of n into parts greater than one, in terms of binomial coefficients.
Theorem 5. Let m be a positive integer.For n 0, By (7), we see that According to Corollary 2, we can write The following results show that the partial sums of QL 0 (n) ± QL 1 (n) can be be expressed as a sum over all the partitions of n in terms of binomial coefficients.The sequence Theorem 6.Let m be a positive integer.For n 0, By Theorem 6, with m replaced by one, we obtain the following identity.
By (6), we see that According to Corollary 3, we can write Theorem 7. Let m be a positive integer.For n 0, The case m = 1 of Theorem 7 reads as follows.
Corollary 4. For n 0, By (7), we see that According to Corollary 4, we can write: The remainder of our paper is organized as follows.In Section 2, we consider the elementary symmetric functions and introduce Lemma 1.This result allows us to provide analytic proofs of Theorems 3-7.In Section 3, we consider the complete homogeneous symmetric functions and introduce Lemma 2. This result allows us to obtain new expressions for the partition function QL 0 (n) ± QL 1 (n), the first differences of QL 0 (n) ± QL 1 (n), and the partial sums of QL 0 (n) ± QL 1 (n) as sums over partitions of n in terms of binomial coefficients.In the last section, we consider a sum over all the partitions of n in order to provide a new expression for the generating function of QL 0 (n) ± QL 1 (n).Finding a combinatorial interpretation in terms of n-color partitions for this sum over all the partitions of n remains an open problem.

Proof of Theorems
The results presented in the previous section follow directly from the following lemma.
where {α n } n 1 is a sequence of non-negative integers.
Proof.It is well known that the elementary symmetric functions [17] e k (x 1 , x 2 , . . ., are characterized by the following formal power series identity in t: Taking into account the elementary symmetric functions, we can write x j k e j (1, . . ., 1 where we have invoked the well-known Cauchy multiplications of power series.By this identity, with x k replaced by q k−1 and z replaced by ±q, we obtain The limiting case n → ∞ of this identity reads as This concludes the proof.

Proof of Theorem 3
Taking into account Lemma 1, with α j replaced by m • j, we can write The proof follows easily by equating the coefficients of q n in this relation.

Proof of Theorem 4
Considering the generating function for the first differences of QL(n), we can write Taking into account we obtain by Lemma 1 with α j replaced by m • a j .The proof follows easily taking into account a 1 = 0.

Proof of Theorem 5
In order to prove Theorem 5, we take into account the following generating function: by Lemma 1 with α 1 = 2m and α j = m • j for j 2.

Proof of Theorem 6
Taking into account the generating function for the partial sums of QL(n), we can write by Lemma 1 with α j = m • b j .

Proof of Theorem 7
Taking into account the generating function for the partial sums of QL 0 (n)

Identities of the Second Kind
In order to derive new formulas involving n-color partitions into distinct parts, we take into account the following lemma.
where {α n } n 1 is a sequence of non-negative integers.

Proof. The complete homogeneous symmetric functions [17]
are characterized by the following formal power series identity in z: Taking into account the complete homogeneous symmetric functions, we can write where we have invoked the well-known Cauchy multiplications of power series.By this identity, with x k replaced by q k−1 and z replaced by ±q, we obtain The limiting case n → ∞ of this identity reads as This concludes the proof.
In analogy with Theorem 3, we introduce Theorems 8 and 9.In Theorem 8, for any positive integer n we denote by c n the number of compositions of n in two parts with at least one even part.It is not difficult to prove that for n odd, n/2 − 1, for n even.
We remark that the sequence Theorem 8. Let m be a positive integer.For n 0, Proof.Taking into account the generating function ( 5), we can write by Lemma 2 with α 2n−1 = m(2n − 1) and α 2n = m • n.This concludes the proof.
The case m = 1 of Theorem 8 provides a new decomposition of QL(n) in terms of binomial coefficients as a sum over all the partitions of n.
Corollary 5.For n 0, We remark that the terms in the right-hand side of th formula given by Theorem 8 are all positive.In addition, the first two factors of the product are always equal to 1 (because c 1 = c 2 = 0).According to Corollary 5, for n = 5 we can write In order to present the following result, for a positive integer n we denote by s n the sum of divisors d of n such that n/d is a power of two.For example, the divisors of six are one, two, three and six.Since 6/3 = 2 1 and 6/6 = 2 0 , we have s 6 = 3 + 6 = 9.We remark that the sequence (s n ) n 1 = (1, 3, 3, 7, 5, 9, 7, 15, 9, 15, 11, 21, 13, 21, 15, 31, 17, 27, . ..) is known and can be seen in the On-Line Encyclopedia of Integer Sequence ([15] A129527).Theorem 9. Let m be a positive integer.For n 0, Proof.Taking into account the generating function (5), Lemma 2, and the identity we can write This concludes the proof.
The case m = 1 provides the following identity.
Corollary 6.For n 0, We remark that the first factor of the product is always equal to 1 (because s 1 − 1 = 0).The case n = 5 of Corollary 6 reads as follows: In analogy with Theorem 4, we have the following result.
Theorem 10.Let m be a positive integer.For n 0, Proof.By the proof of Theorem 8, we see that Taking into account Lemma 2, we can write This concludes the proof.

Corollary 7.
For n 0, We remark that the first factor of the product is always equal to 1 (because c 2 = 0).The case n = 5 of Corollary 7 reads as follows: In analogy with Theorem 5, we have the following result, where {r n } n 1 is a sequence of positive integers defined as follows: Theorem 11.Let m be a positive integer.For n 0, Proof.By the proof of Theorem 9, we see that Taking into account the generating function (5), the identity (10), and Lemma 2, we can write This concludes the proof.
Corollary 8.For n 0, The case n = 5 of Corollary 8 reads as follows: In analogy with Theorem 6, we have the following result.
Theorem 12. Let m be a positive integer.For n 0, Proof.Taking into account the generating function (12) and Lemma 2, we can write This concludes the proof.
The case n = 5 of Corollary 9 reads as follows: In analogy with Theorem 7, we have the following result, where {v n } n 2 is a sequence of positive integers defined as follows: Theorem 13.Let m be a positive integer.For n 0, Proof.Taking into account the identity (10), the generating function (13), and Lemma 2, we can write This concludes the proof.

Concluding Remarks
A collection of identities involving QL 0 (n) ± QL 1 (n) has been introduced in this paper by considering specializations of complete and elementary symmetric functions.In this context, as consequences of these results, we remark some identities involving binomial coefficients.
The following identity follows from Theorems 3 and 8.
Corollary 11.Let m be a positive integer.For n 0, The following identity follows from Theorems 3 and 9.
By this theorem, with t replaced by q, we obtain a well-known expression for the generating function of Q(n), the number of partitions of n into distinct parts, i.e., ∞ ∏ n=1 (1 + q n ) = ∞ ∑ n=0 q ( n 2 )+n (q; q) n .
We remark an analogous result for the generating function of QL(n).