k-complementing subsets of nonnegative integers

A collection { S 1 , S 2 , … } of nonempty sets is called a complementing system of subsets for a set X of nonnegative integers if every element of X can be uniquely expressed as a sum of elements of the sets S 1 , S 2 , … We present a complete characterization of all complementing systems of subsets 
for the set of the first n nonnegative integers as well as an explicit enumeration formula.


Introduction
Let S = {S 1 ,S 2 ,...} represent a collection of nonempty sets of nonnegative integers in which each member contains the integer 0. Then S is called a complementing system of subsets for X ⊆ {0, 1,...} if every x ∈ X can be uniquely represented as x = s 1 + s 2 + ••• with s i ∈ S i .We will also write X = S 1 ⊕ S 2 ⊕ ••• and, when necessary, refer to X as the direct sum of the S i .
We will denote the set of all complementing systems for X by CS(X).Then {X} ∈ CS(X) = ∅.
If there is a positive integer k such that X = S 1 ⊕ ••• ⊕ S k , then {S 1 ,...,S k } will be called a k-complementing system of subsets, or a complementing k-tuple, for X.
Denote the set of all complementing k-tuples for X by CS(k,X).We will address the problem of characterizing all S ∈ CS(k,N n ), where N n = {0, 1,..., n − 1}.The corresponding more general problem for CS(N) was solved by de Bruijn [2], where N = {0, 1,...}.Long [4] has given a complete solution for CS(2, N n ).Since the appearance of Long's paper, no progress seems to have been made to solve the problem for k > 2. Tijdeman [6] gives a survey of the evolution of this problem and related work.
In Section 2, we give an alternative proof of Long's theorem (Theorem 2.5) followed in Section 3 by its natural extension (Theorem 3.2) and a general structure theorem for CS(k,N n ) (Theorem 3.5).
A complementing system S = {S 1 ,S 2 ,...} ∈ CS(N) will be called usual if for any sequence g 1 ,g 2 ,... (g i > 1) of integers, each S i ∈ S is given by where m 0 = 1, m i = g 1 g 2 ••• g i (i > 0).We will refer to the collection {S 1 ,S 2 ,...,S i ,...} as the complementing system corresponding to (or generated by) the integers g 1 ,g 2 ,....We will denote the set of all usual complementing systems of subsets for N by UCS(N).For positive integers a and c, the set U = {0, a,2a,...,(c − 1)a} will be called a simplex, written additively (after Tijdeman [6]).We will adopt the notation U = [a,c].Thus, by (1.1) every member of a usual complementing system is a simplex.
We can derive usual complementing systems of subsets for N n from the following adaptation of a theorem of Long [4].
) represent any factorization of n as a product of positive integers and let the sets S 1 ,...,S k be defined as in (1.1).Then to form a complementing pair for N n , the least nonzero element of S 2 must be g 1 (since S 1 already contains 0,1,...,g 1 − 1) and thenceforth elements of S 2 must be consecutively spaced g 1 apart.This shows that S 2 has the form S 2 = [g 1 ,g 2 ].Assume that the proposition holds for some fixed integer v and consider the system {S 1 ,...,S v ,S v+1 }.By the inductive hypothesis, ,S v+1 }; and the case for k = 2 shows that S v+1 has the required form.Hence the theorem is proved by mathematical induction.
(ii) It is clear that Usual complementing systems for finite sets will be taken to include all of the systems aS = {aS 1 ,...,aS k } ∈ UCS(aN n ) (a ≥ 1).
(iii) It follows from Theorem 1.
, where f (n) denotes the number of ordered factorizations of n.A simple bijection is as follows: given any ordered f (n) can be computed using the recurrence [5,7] where f (1) = 1 and the sum is over divisors d of n, d < n.
If n has the prime factorization , then f (n) can also be found using MacMahon's formula [7]: where We deduce at once that | UCS(k,N n )| = f (n,k), where UCS(k,N n ) denotes the set of usual k-complementing systems of subsets for N n and f (n,k) is the number of ordered k-factorizations of n.
It follows from (1.3) (see also [1, page 59]) that f (n,k) can be computed from the formula .. belong to a certain class of p. de Bruijn [2] is followed and T is called a degeneration of S. When necessary, T is also said to be induced by the partition or shape p, without reference to S.
The following fundamental classification theorem [2] for complementing systems of subsets for N, which also applies to N n via Theorem 1.1, is crucial to all what follows.Theorem 1.4 (N.G. de Bruijn).Every complementing system of subsets for N is the degeneration of a usual complementing system.
Other relevant properties of usual complementing systems are summarized in the next theorem.
Theorem 1.5.(i) A collection of sets S is a usual complementing system for a finite set if and only if S ∈ CS(X), where X is a simplex. (ii is the direct sum of more than one simplex if and only if n is composite. (iv) For S ∈ UCS(N) to be the degeneration of T ∈ UCS(N), where S = T, it is necessary and sufficient that some member of S has composite cardinality.
Proof.(i) If S is a usual complementing system for a finite set, then S is generated by a finite sequence of positive integers.Thus by Theorem 1.1 and Remark 1.
, where a and n are positive integers, then by Remark 1.2(ii) we can form (ii) This follows from Remark 1.2(i) and part (i).
Remark and Definition 1.6.Theorem 1.5(iv) implies that a fixed P ∈ CS(N) is not the nontrivial degeneration of any T ∈ UCS(N) if and only if each P i ∈ P has prime cardinality, that is, P is a usual complementing system generated by a sequence of prime numbers.P will be called a prime complementing system of subsets for N.
Hence we have the following.
Corollary 1.7.Every usual complementing system of subsets is a degeneration of a prime complementing system.
We can now state the following complementing subset-systems analogue of the fundamental theorem of arithmetic.
Theorem 1.8.Every complementing system of subsets is a degeneration of a prime complementing system.
Proof.The theorem follows by transitivity from Theorem 1.4 and Corollary 1.7.
Remarks 1.9.(i) Denote the set of prime complementing systems for N by PCS(N).It is clear that there are strict inclusions: PCS(N) ⊂ UCS(N) ⊂ CS(N).
(ii) Theorem 1.8 guarantees that to generate all complementing systems via degenerations it suffices to use the minimal generating set PCS(N) rather than the whole UCS(N).The same remark applies to the finite case for the corresponding sets PCS(N n ), CS(N n ), and  Remark 2.2.A close observation of Table 2.1 shows that the nc(m,k) are just Stirling numbers of the second kind which have been shifted one step to the right and one step down, that is,  Thus Assume that (2.3) holds for all positive integers up to m.Then Theorem 2.1 gives where the second equality follows from the inductive hypothesis and the last equality follows from the usual recurrence for s2(m,k).Thus (2.3) is also established by mathematical induction.
Hence the standard formula [3, page 251] yields the following corresponding formula: If b * (m) denotes the total number of nonconsecutive partitions of {1, 2,...,m}, then it is easily deduced from where B(m) denotes the mth Bell number.
Proof.This follows from (2.6) or, more completely, from the proof of Theorem 2.1.
Notation 2.4.Given any S ∈ CS(N n ), let Degen(S) denote the set of all degenerations of S. Let Degen(S,k) denote the set of all k-degenerations of S and let degen(S,k) be an element of Degen(S,k).Theorem 1.4 says that CS(N n ) = ∪(Degen(S), S ∈ UCS(N n )), and implies that CS(k,N n ) = ∪(Degen(S, k), S ∈ UCS(N n ) and |S| ≥ k).
It is now straightforward to deduce the following characterization theorem [4, Theorems 1 and 2] for complementing pairs for N n .
Theorem 2.5 (C.T. Long).(i) {A, B} ∈ CS(2,N n ) (n ≥ 2) if and only if there exists a sequence g 1 ,g 2 ,...,g v of integers corresponding to the factorization n = g 1 •••g v of n such that A and B are sets of all finite sums of the form Proof.(i) This follows from the fact that the unique shape p = {{1, 3,...},{2,4,...}} given by Lemma 2.3 induces degen(S,2) Remark 2.6.In his original theorem, Long [4] states the result of Theorem 2.5(ii) as However, the strict form given above is more suitable for generalization as shown below.

Essential complementing k-tuples and a structure theorem
will be called essential if it is induced by the partition p of {1, 2,...,v} into a complete set of residue classes, modulo k. p is also referred to as essential.
Denote the set of essential k-complementing systems of subsets for We have the following natural extension of Long's theorem.
if and only if there exists a sequence g 1 ,g 2 ,...,g v of integers corresponding to the factorization n = g 1 ••• g v (k ≤ v) of n such that each set T i consists of all finite sums of the form where Proof.(i) For each k and essential partition p of {1, ...,v | v ≥ k} there is an injective degeneration map dgn(p) : Hence dgn(p) is a well-defined mapping.The injectivity of dgn(p) is easily established by following the above implications backward (see also the statement immediately following (3.4) below).The image of dgn(p) is clearly ECS(k,N n ).We see that the restriction of dgn(p) to UCS(k,N ) is the identity map.
(ii) By part (i) and Remark 1.2(iii) we have We next define the class vector of a set partition [8].
Definitions 3.3.The class vector of a k-partition of {1, ...,v} is the v-vector (e 1 ,...,e v ) in which e i ∈ {1, ...,k} and e i belongs to class i for each i.
Thus for 1 ≤ k ≤ w ≤ v, the contraction map can be defined by setting F(q) = q if q ∈ NC(w,k) and F(q) = p if p is represented by the class vector obtained from the class vector h q of q by replacing every sequence of equal and consecutive components e i ,e i+1 ,...,e i+c ∈ h q with the common value e i .It follows that the restriction of F to NC(w,k) is the identity map.
We now turn to the problem of characterizing all complementing k-tuples for N n .First we observe that it is not possible to state a simple rule for all T ∈ CS(k,N n ), k > 2, as appeared in (3.1) since the sets in a general p ∈ SP(v,k) (v > k) can be constituted quite arbitrarily.
Theorem 1.4 implies that every T ∈ CS(k,N n ) is induced by some p ∈ SP(v,k), k ≤ v.But operationally we need only NC(v,k), and not SP(v,k), to determine all of CS(k,N n ), using (3.2), in view of the surjective contraction map (3.4) and the fact that Theorem 1.5(ii) enables the automatic coupling of consecutive simplices thus making all partitions in SP(v,k) − NC(v,k) redundant.Hence the partitions in NC(v,k) effectively account for all S ∈ CS(k,N n ), for each v ≥ k.
Since nc(k,k) = 1 and the singleton NC(k,k) contains the essential partition, it follows from (3.2) that every map dgn(q) in which q ∈ NC(v,k) is not the essential partition is necessarily defined on the reduced domain {S ∈ UCS(N n ) | |S| > k}.Hence for each v (1 ≤ v ≤ Ω(n)), there exist precisely nc(v,k) maps, dgn(p), with p ∈ NC(v,k); and so by Remark 1.2(iii), there is a total of f (n,v)nc(v,k) contributions to CS(k,N n ).Thus | CS(k,N n )| may be found by summing f (n,v)nc(v,k) over v.
Hence we obtain the following structure theorem for CS(k,N n ).
Theorem 3.5.(i) {T 1 ,T 2 ,...,T k } ∈ CS(k,N n ) (n ≥ 2) if and only if there exists a sequence g 1 ,g 2 ,...,g v of integers corresponding to the factorization n = g 1 ••• g v of n such that each set T i is given by all finite sums of the form ) where the second equality follows from (2.3) and f (n,k) denotes the number of ordered kfactorizations of n.
Remark 3.6.(i) Theorem 3.2 follows from Theorem 3.5 by noting that the essential partition p is the unique member of NC(v,k) such that dgn(p) is defined on the maximal domain {S ∈ UCS(N n ) | |S| ≥ k} which forces nc(v,k) = 1 for all v ≥ k.
(ii) We observe that any fixed S ∈ UCS(k,N n ) gives rise, via degenerations, to a total of B(k) complementing systems T ∈ CS(N n ).In particular, if n is a prime power, then a single Bell number counts the whole of CS(N n ) in view of Theorem 1.8, that is, ( (iv) Thus the function B(m) − 2 m−1 , m = 1,2,... [5] also counts the complementing systems of subsets for {0, 1,..., p m − 1} in which at least one member is not a simplex or, equivalently, the partitions of the set {1, 2,...,m} in which at least one class of each partition contains a pair of nonconsecutive integers.

2 .
Complementing pairs and the theorem of C. T. Long Let SP(m) denote the set of all partitions of {1, 2,...,m} so that | SP(m)| = B(m), the mth Bell number.Also let SP(m,k) denote the set of all k-partitions of {1, 2,...,m} so that | SP(m,k)| = s2(m,k), a Stirling number of the second kind.An element p of SP(m,k) will be called nonconsecutive if no member of p contains a pair of consecutive integers.Let NC(m,k) denote the set of all nonconsecutive kpartitions of {1, 2,...,m}, and let nc(m,k) = |NC(m, k)|.Theorem 2.1.nc(m,k) satisfies the following recurrence:

. 1 )
Proof.To find a p ∈ NC(m,k) (m > k > 2), we can either insert the singleton {m} into any p ∈ NC(m − 1,k − 1) or put the integer m into any k − 1 members of a p ∈ NC(m − 1,k) which do not contain m − 1.There are clearly (k − 1)nc(m − 1,k) possibilities in the second case.Hence the main result follows.The boundary conditions are clear from the definition and imply that nc(m,1) = 0 (m = 1), nc(m,m) = 1.
, 2,...,v}, and NC(v,k) is the set of all k-partitions of {1, 2,...,v} in which no member of each partition contains a pair of consecutive integers.