On k-partitions of multisets with equal sums

We study the number of ordered k-partitions of a multiset with equal sums, having elements α1,…,αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1,\ldots ,\alpha _n$$\end{document} and multiplicities m1,…,mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1,\ldots ,m_n$$\end{document}. Denoting this number by Sk(α1,…,αn;m1,…,mn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k(\alpha _1, \ldots , \alpha _n; m_1, \ldots , m_n)$$\end{document}, we find the generating function, derive an integral formula, and illustrate the results by numerical examples. The special case involving the set {1,⋯,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,\dots ,n\}$$\end{document} presents particular interest and leads to the new integer sequences Sk(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k(n)$$\end{document}, Qk(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k(n)$$\end{document}, and Rk(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k(n)$$\end{document}, for which we provide explicit formulae and combinatorial interpretations. Conjectures in connection to some superelliptic Diophantine equations and an asymptotic formula are also discussed. The results extend previous work concerning 2- and 3-partitions of multisets.

introduced numerous concepts which are still in use, and proved fundamental results concerning partitions of an integer n into distinct parts, or into odd parts.
Since that time, numerous mathematicians including Gauss, Cauchy, Jacobi, Weierstrass, MacMahon, Hardy, Ramanujan, Erdös, and Andrews have contributed to the study of partitions. Key details about the history of partitions can be found in the classical books of Andrews [1] and [2], or in the review article by Pak [17], which focuses on bijective proofs of classical partitions identities.
First noticed by Euler, the number p(n) of ways in which a positive integer n can be partitioned as a sum of positive integers is given by the generating function: For details about generating functions see [21]. The asymptotic estimation for p(n) was first given by Hardy and Ramanujan, who in 1918 proposed the formula: valid as n → ∞. An elementary proof was given in 1942 by Erdös [14].
Of particular interest is the study of partitions of the set [n] = {1, 2, . . . , n} into two sets with equal sums, which gives rise to the signum equation. For a positive integer n, the number of solutions is denoted by S(n) and corresponds to the number of ways in which one can choose + and − such that ±1 ± 2 ± 3 ± · · · ± n = 0. This sequence is indexed as A063865 [18] and was shown to be closely related to the Erdös-Surányi problem, by Andrica and Ionaşcu [5]. The asymptotic formula for S(n) was conjectured in 2002 by Andrica and Tomescu [6]: This was proved in 2013 by Sullivan [20], using analytic methods. Proof details and possible extensions based on the Central Limit Theorem, as well as connections to Erdös-Surányi representations of integers, have been suggested [4,5]. Another important and challenging problem is to enumerate the partitions of multisets having certain properties. Results for the number of partitions of multisets, as well as asymptotic formulae for small multiplicities were obtained in the 1970's by Bender [12], and Bender et al. [13]. Further information about multiset theory can be found in the book by Stanley [19].
The current paper is motivated by initial results concerning the number of 2partitions with equal sums for multisets [11] and by the more recent extension to 3-partitions with equal sums, in which different techniques were developed [7]. The framework for the general study of k-partitions with equal sums for multisets is inspired from the generating function approach proposed by Andrica in [3], which started initially from a problem involving derivatives. This opened the way to numerous novel results related to multi-partitions with equal sums.
The structure of the paper is described below. In Sect. 2, we introduce the number S k (α 1 , . . . , α n ; m 1 , . . . , m n ), of ordered k-partitions of a multiset having equal sums, for which we obtain the generating function and an integral formula. The case of k-partitions of the set [n] = {1, . . . , n} is discussed in Sect. 3. The main results in Sect. 4 are contained in Theorems 1 and 2. This leads to the arithmetic functions S k (n), Q k (n), and R k (n), for which we find explicit formulae, combinatorial interpretations, novel integer sequences, and conjectures related to superelliptic equations and to the asymptotic formula.

k-Partitions of multisets with equal sums
One of the famous strongly NP-complete problems is the 3-partition problem, which has the following statement. Given the positive integers b, m and a 1 , . . . , a n such that n = 3m and n s=1 a s = mb, one needs to partition the set {a 1 , . . . , a n } into m subsets, each containing exactly three elements, in which sum is exactly b (see, e.g., [15] and [16]  Definition 1 Let k ≥ 2 be an integer. Denote by S k (α; m) the number of ordered k-partitions of M having equal sums, i.e., the number of k-tuples (C 1 , . . . , C k ) of pairwise disjoint subsets of M such that The number S k (α; m) is the free (constant) term of the expansion of the Laurent polynomial F(X 1 , . . . , Indeed, assume that for s = 1, . . . , n, from X α s 1 + · · · + X α s k−1 + This is equivalent to where P m, j (X 1 , . . . , X j−1 , X j+1 , . . . , X k−1 ) are Laurent polynomials. As F is symmetric in its variables, the Laurent polynomials P m, j do not actually depend on j; hence, we may use the simplified notation P m (X 1 , . . . , X j−1 , X j+1 , . . . , X k−1 ). One can easily check that the free term of F(X 1 , . . . , X k−1 ) is equal to the free term of Let X j = cos t + i sin t in (3) and (5) and integrate with respect to t over the range [0, 2π ]. Since the integral of the monomial X m j with respect to t over [0, 2π ] vanishes for m = 0, the integral representation of the polynomial is given by Setting X j = X , X l = 1 for l = 1, . . . , k − 1 and l = j in (3) and (5), one obtains By symmetry in X and X −1 , we have Also, from (6), we deduce that which depends on k, α, and m. In what follows, we shall denote for simplicity We have that where R k (m; α) is the sum of the coefficients of P 0 ( X j ) that are different from the free term. Also, setting X = 1 in (7), we get k m 1 +···+m n = m∈Z P m (1 . . . , 1), that is the sum of all the coefficients in all polynomials P m is k m 1 +···+m n . By formula (9), after simple computations, it follows that As shown in [11], the integral formula for the number of ordered 2-partitions with equal sum, of the multiset M with integer elements α and multiplicities m, is given by For α s = s and m 1 = · · · = m n = m, s = 1, . . . , n, this produces the formula: where c

k-Partitions with equal sums of the set [n] = {1, . . . , n}
In this section, we set α s = s and m s = 1, for s = 1, . . . , n, and use the simplified notations S k (n) for S k (α; m) and R k (n) for R k (α; m). Some recurrences can be obtained for the coefficients of the polynomial F(X 1 , . . . , X k−1 ) defined by (3), which is indexed by the level n, as in the formula: We may first write F n (X 1 , . . . , X k−1 ) as a Laurent polynomial in X 1 , . . . , X k−1 with integer coefficients, given by the formula: Clearly, we have where U and V are polynomials. If V = 0 at the origin of R k−1 , then the coefficients in (14) are given by the Cauchy integral formula: with T a product of sufficiently small circles around the coordinate axes of R k−1 .
A detailed derivation of this approach, further applications and examples, can be found in [8,Chapter 4.3] and the references therein. While this is a closed formula, for effective computations, it is sometimes more practical to use a recurrence between the coefficients of levels n and n − 1 (see Theorems 4 and 5 in [9]).
From the integral formula (9) applied in this case, we can also compute the terms P n,0 directly. Since this computation is actually performed for a given value of k, for what follows, we introduce the notation: Simple calculations show that and by (9) and (12) it follows that Q 2 (n) = c (1) 0 (n).

Main results concerning Q k (n)
In this section, we explore properties and special cases for Q k (n), S k (n), and R k (n). In the process, we provide new context for a number of existing sequences and identify novel entries to [18]. We then establish a polynomial formula for Q k (n) and conjecture its asymptotic behavior. We also conjecture properties related to the distribution of perfect powers within this sequence. Denoting by T (n, k) the number of set partitions of [n] = {1, . . . , n} into k blocks with equal sum of elements, indexed as A275714 in [18], the number of ordered partitions satisfies S k (n) = k! · T (n, k) and generates a new sequence.
The following result provides a new combinatorial context for Q k (n). (13)    and is indexed as A007576. These terms can also obtained by formula (18) as

Theorem 1 Q k (n) is the number of ordered partitions of [n] into k disjoint sets
It was recently conjectured (see context for A007576 [18]) that As the monomials in the expansion of F n (X , Y ) are of the form X α Y β (XY ) −γ , a term is independent of X if and only if α = γ . To identify the numbers of terms independent of X , we need to count all subsets A and B of {1, 2, . . . , n} such that ). This is equivalent to finding the number of solutions of the 3-signum equation (20). For k = 3, we list below the triples obtained for n = 4 and n = 5.

Remark 1 In
This result will be fully explained in general, in the following subsection.

The polynomial formula for Q k (n)
We show that Q k (n) can be expanded as a polynomial in the variable k − 2, and we calculate its coefficients. We prove that these are non-negative integers for which we can establish a number of properties.

is the number of ordered partitions of [n] into 3 subsets A, B, C, such that the cardinality of B is d and σ (A) = σ (C). 2 • The coefficients
where σ d (cos t, cos 2t, . . . , cos nt) is the d-th symmetric sum of cos t, cos 2t, . . . , cos nt.
Proof Recall that by formula (18), we have Consider a 3-partition A, B, C of [n] such that σ (A) = σ (C) and assume that A∪C has d = 0, . . . , n elements (denote the number of such 3-partitions by N (d, n)). The number of ordered partitions of [n] into k subsets generated by this configuration, having the property A 1 = A, B = A 2 ∪ · · · ∪ A k−1 , and A k = C is (k − 2) n−d (i.e., the number of functions between a set with n − d elements and a set with k − 2 elements). It is now clear that Q k (n) is a linear combination of powers of k − 2, where the coefficient of (k − 2) n−d has the multiplicity N (d, n). This phenomenon is illustrated in Tables 1, 2, and 3. 2 • For d = 0, . . . , n, the coefficients are hence the result. N k 1 ,...,k d be the number of solutions of the 2-signum equation The following relation holds true:

Corollary 1 Let
Proof The following identity is known to hold (see, e.g., [4]): where the sum is taken over all the choices of + and −. Also, notice that for an integer k ∈ Z, one has 1 2π 2π 0 cos kt dt = 1, k = 0 0, k = 0.
Summing over all such configurations, we obtain (26).

Some results and conjectures concerning Q k (n)
By (23), the coefficients of Q k (n) are non-negative integers, in which combinatorial meaning is related to the partitions of the set [n]. We shall present arithmetic properties of Q k (n), highlight connections to hyperelliptic equations, and formulate conjectures related to elliptic curves and to the asymptotic behavior of Q k (n). Direct calculations involving the integral (18) lead to the formulae: The sequence Q k (3) recovers A034324 in [18], while the sequences Q k (n) obtained for n = 4, . . . , 9 are new and have interesting arithmetic properties.
It has been claimed that the sequence Q k (3) A084380 does not contain any perfect squares, i.e., the elliptic equation This statement is linked to a Catalan-type conjecture related to Pillai's equation a x − b y = n, with a > 0, b > 0, x > 1, y > 1 integers, which states that for any integer n, there are finitely many perfect powers in which difference is n. For n = 2, the only solution involving perfect powers smaller than 10 18 was 2 = 3 3 − 5 2 . The number of such solutions is linked to A076427 in [18].
We now formulate the following conjectures.

Conjecture 1
The sequence Q k (4) does not contain any perfect cubes, i.e., the superelliptic equation x 4 + 4x + 2 = y 3 has no solution in positive integers.
On the other hand, it can be easily seen that the sequence Q k (4) does not contain perfect squares, since for x ≥ 3, we have (x 2 ) 2 < x 4 + 4x + 2 < (x 2 + 1) 2 .

Conjecture 2
The sequence Q k (6) does not contain any perfect fifth powers, i.e., the superelliptic equation x 6 + 12x 3 + 16x 2 + 6x = y 5 has no solution in non-negative integers, apart from the trivial solution x = y = 0.

Conjecture 4
For k ≥ 2, the following asymptotic formula holds: as n → ∞.
The formula for Q 2 (n), conjectured by Andrica and Tomescu in 2002, was proved by Sullivan in 2013. This formula is also consistent with the asymptotic evaluations conjectured for Q k (3) in Example 4, and for Q k (4) in Example 5 (Fig. 1).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.