Some remarks on 3-partitions of multisets

Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m1, ...,mn;α1, ..., αn), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.


Introduction
The signum equation for a given sequence of integers is considered in [3], in connection with the Erdös-Surányi problem. In particular, for a given integer n ≥ 2, the level n solution of this equation represents the number S(n) of ways of choosing + and − such that ±1 ± 2 ± 3 ± · · · ± n = 0. This is also the number of ordered partitions of {1, 2, . . . , n} in two sets with equal sums.
Andrica and Tomescu [4] conjectured an asymptotic formula for S(n): which was proved by analytic methods by Sullivan [11].
Starting from a problem involving derivatives, Andrica established a generating function which allowed novel approaches in the study of 2-partitions with equal sums for multisets [1]. We refer the reader to [2,3] for connections with Erdös-Suranyi representations, to [10] for general theory of multisets and to [12] for details about generating functions. This paper is motivated by some recent results on the number of ordered 2-partitions with equal sums for multisets obtained in [5]. The study of 3partitions of multisets differs essentially from that of 2-partitions. In Section 2 of this paper we investigate the number of ordered 3-partitions of a multiset M having equal sums, for which establish the generating function and a useful integral formula. Some particular instances related to the number of solutions for the 3-signum equation are studied in Section 3, where recurrence formulae are established and some new integer sequences are proposed.

3-partitions of multisets with equal sums
Partitions have direct applications to classical combinatorial optimization problems such as Bin Packing Problem (BPP), Multiprocessor Scheduling Problem (MSP) and the 0-1 Multiple Knapsack Problem (MKP) [6].
Of particular interest is the 3-partition problem, one of the famous strongly NP-complete problems [7,8]. Given a positive integer b and a set [n] = {1, 2, ..., n} of n = 3m elements, each having a positive integer size a s , such that n s=1 a s = mb. The problem has a solution if there is a partition of N into m subsets, each containing exactly three elements from N, whose sum is exactly b. For example, the set {10, 13, 5, 15, 7, 10} can be partitioned into the two sets {10, 13, 7}, {5, 15, 10}, each of which sum to 30.

}.
We call m s the multiplicity of the element α s in the multiset M, while the notation σ(M) = n s=1 m s α s represents the sum of the elements of M. Definition 2.1 Denote by S(m 1 , ..., m n ; α 1 , ..., α n ) the number of ordered 3partitions of M having equal sums, i.e., the number of triplets ( Indeed, assume that in the product X αs + Y αs + 1 (XY ) αs ms we have selected c s 1 terms equal to X αs , c s 2 terms equal to Y αs , and c s 3 terms equal to 1 (XY ) αs , with s = 1, . . . , n, and notice that in this case we must have c s 1 + c s 2 + c s 3 = m s . Such a selection contributes to the free term if and only if This means that the sets , · · · , α n , · · · , α n c j n times }, j = 1, 2, 3, represent a partition of M which also satisfies property (ii) in Definition 2.1.
Ordering (1) in the increasing order of integer powers, one can write where P m (Y ) and Q m (X) are Laurent polynomials. Also, notice that the free term of F (X, Y ) is the free term of P 0 (Y ) and Q 0 (X). Clearly, we can write Considering X = cos t + i sin t, in (3) and integrating with respect to t over the interval [0, 2π], one obtains the following integral representation of the polynomial Setting Y = 1 in (3) one obtains which by symmetry in X and X −1 gives that Also, from (4) we deduce that Since X αs + 1 + 1 X αs = 1 + 2 cos α s t, we have P 0 (1) = 1 2π (1 + 2 cos α s t) ms dt.
Proof. The following formula can be established.
As the monomials in the F n (X, Y ) expansion have the form X α Y β (XY ) −γ , a term is independent of X if and only if α = γ. For a given n, we have to enumerate all the partitions A, B, C of [n] having the property σ(A) = σ(C). The problem is equivalent to finding all triplets (α, β, γ) such that α, β, γ ≥ 0, α = γ and α + β + γ = σ([n]).