Counting numerical sets with no small atoms

A numerical set S with Frobenius number g is a set of integers with min(S) = 0 and max(Z − S) = g, and its atom monoid is A(S) = {n ∈ Z | n+ s ∈ S for all s ∈ S}. Let γg be the number of numerical sets S having A(S) = {0} ∪ (g,∞) divided by the total number of numerical sets with Frobenius number g. We show that the sequence {γg} is decreasing and converges to a number γ∞ ≈ .4844 (with accuracy to within .0050). We also examine the singularities of the generating function for {γg}. Parallel results are obtained for the ratio γ σ g of the number of symmetric numerical sets S with A(S) = {0}∪(g,∞) by the number of symmetric numerical sets with Frobenius number g. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases. Let Z denote the additive group of integers and let N denote the monoid of nonnegative integers. Both of these sets are linearly ordered by the Archimedean ordering and we will use standard interval notation to describe their convex subsets. If n ∈ Z and S ⊆ Z then the translate of S by n is the set n+ S = {n+ s | s ∈ S}. A numerical set S is a cofinite subset of N which contains 0, and its Frobenius number is the maximal element in the complement N−S. Equivalently, a numerical set S with Frobenius number g is a set of integers with min(S) = 0 and max(Z−S) = g. A numerical set which is closed under addition is called a numerical monoid. Every numerical set S has an associated atom monoid A(S) defined by A(S) = {n ∈ Z | n+ S ⊆ S} , and this is easily seen to be a numerical monoid with the same Frobenius number as S. Note that A(S) ⊆ S and that S is a numerical monoid if and only if A(S) = S. The nonzero elements of A(S) are referred to as the atoms of S. For each g ≥ 0 let Ng be the numerical monoid Ng = N− [1, g] = {0} ∪ (g,∞) , which has Frobenius number g when g > 0. The atom monoid of every numerical set S with Frobenius number g contains Ng and the complement S−Ng is a subset of (0, g). Conversely, the union of Ng with any subset of (0, g) is a numerical set with Frobenius number g. Therefore the set S(g) = {S ⊆ N | S is a numerical set with Frobenius number g} 1This definition differs from that employed in [AM] where a ‘numerical set’ would be a translate n + S of a numerical set S (in the sense given here) by an arbitrary integer n. Since the atom monoid of n+ S equals the atom monoid of S, this variation of the definition should not lead to any confusion. 2The Frobenius number of N0 = N is −1, and this is the only numerical set with nonpositive Frobenius number. 1 ar X iv :0 80 5. 34 93 v1 [ m at h. C O ] 2 2 M ay 2 00 8 2 JEREMY MARZUOLA AND ANDY MILLER is in one-to-one correspondence with the power set P(0, g) consisting of all subsets of (0, g), and S(g) has cardinality 2g−1. The subset of S(g) consisting of numerical monoids is a much more difficult set to enumerate. This is examined in Backelin’s paper [B] where it is shown that for large values of g roughly 3× 2b(g−1)/2c of the 2g−1 elements of S(g) are numerical monoids. If M ∈ S(g) is a numerical monoid then the anti-atom set of M is the set G(M) = {S ∈ S(g) | A(S) = M} . This is contained in the larger set S(M) = {S ∈ S(g) |M ⊆ A(S)} whose elements might be considered to be ‘M -modules’. Notice that S(g) = S(Ng) and we will also write G(g) = G(Ng). This paper is motivated by the following question which we shall refer to as the Anti-Atom Problem. For a given numerical monoid M with Frobenius number g how many numerical sets in S(g) have atom monoid M? Thus, for a given monoid M , the Anti-Atom Problem asks to compute the cardinality of G(M). This problem is certainly unwieldy given that it fundamentally presupposes an enumeration of the set of numerical monoids in S(g)—an enumeration which Backelin has shown to be intractable at best. Nevertheless we will be able to frame aspects of the problem in a clearer light. Our main result will show that there is one monoid M in S(g) (that monoid being M = Ng) which itself is the atom monoid for approximately 48.4% of all numerical sets in S(g) for large values of g. In order to describe this in more depth we first need to discuss symmetry and pseudosymmetry in numerical sets. These concepts are important throughout much of the theory of numerical monoids and numerical sets (see [FGH], [AM] and [A] for example), and will play a role in many of our discussions. A numerical set S ∈ S(g) is symmetric if an integer x is an element of S if and only if g − x is not an element of S. In other words, S is symmetric when the reflection on Z given by x 7→ g − x carries S onto its complement Z − S. Notice that only numerical semigroups with odd Frobenius number can be symmetric. A numerical set with even Frobenius number g is said to be pseudosymmetric if g/2 / ∈ S and for each integer x 6= g/2, x is an element of S if and only if g − x is not an element of S. Symmetry and pseudosymmetry can also be described using the notion of duality of numerical sets. If S ∈ S(g) then the dual of S is the numerical set S∗ = {n ∈ Z | g − n / ∈ S}, and it is not hard to show that S∗ ∈ S(g) and that A(S∗) = A(S) (more background can be found in section 1 of [AM]). The numerical set S is symmetric if and only if S∗ = S, and it is pseudosymmetric if and only if g is even and S∗ = S ∪ {g/2}. For each numerical set S ∈ S(g) there is a rational number type(S) no smaller than one, called the ‘type of S’, which satisfies the property that S is symmetric if and only if type(S) = 1. The type of a numerical monoid M ∈ S(g) is always an integer, and it equals the cardinality of its omitted atom set O(M) = { n ∈ Z−M | n+ ( M − {0} ) ⊆M } . Since O(M) ⊂ N and g ∈ O(M), the type of a numerical monoid M ∈ S(g) is in the interval [1, g], and the largest possible value type(M) = g is only achieved when M = Ng. The 3In [BF] the elements of S(M) are called ‘relative ideals over M ’. 4 More generally, if the symmetric difference of S and S∗ contains no more than one element then S is symmetric, pseudosymmetric or “dually pseudosymmetric” (meaning that S∗ is pseudosymmetric). COUNTING NUMERICAL SETS WITH NO SMALL ATOMS 3 following elementary results allow us to solve the Anti-Atom Problem for symmetric and pseudosymmetric numerical monoids. Proposition 1. Suppose that M is a numerical monoid and that S is a numerical set with A(S) = M . Then M ⊆ S ⊆M∗. Proof. Let S be a numerical set in S(g) with A(S) = M and s ∈ S. If g − s were an element of M then g = s+ (g − s) would be an element of S, which contradicts g being the Frobenius number of S. Thus g − s / ∈ M which implies that s ∈ M∗, and M = A(S) ⊆ S ⊆M∗. Corollary 2. A numerical monoid M ∈ S(g) is symmetric if and only if there is just one numerical set (which must be M itself) whose atom monoid is M . If M is a pseudosymmetric numerical monoid then there are precisely two numerical sets (which must be M and M∗) whose atom monoid is M . Proof. Let M ∈ S(g) be a monoid. If M is not symmetric then M 6= M∗ but A(M∗) = A(M) = M , and so there are at least two distinct numerical sets in G(M). On the other hand, if M is symmetric and S ∈ G(M) then M ⊆ S ⊆M∗ = M and S = M . If M is pseudosymmetric and S ∈ G(M) then M ⊆ S ⊆M∗ = M ∪{g/2}, so that S equals M or M∗. This corollary then provides the first positive answers to the Anti-Atom Problem: namely, that |G(M)| = 1 when M is symmetric and that |G(M)| = 2 when M is pseudosymmetric. At the other end of the spectrum, we shall show that the anti-atom set of Ng (which is the numerical monoid in S(g) farthest removed from being symmetric since type(Ng) = g is the largest possible type among all monoids in S(g)) is an order of magnitude larger in size than that of any other numerical monoid with Frobenius number g. To establish this we will examine the sequence γg = |G(g)|/|S(g)|. We introduce a combinatorially defined sequence of positive integers {Ak} with the property that 1−γg is a partial sum of the convergent infinite series ∑∞ k=1 Ak4 −k. This allows us to show that {γg} is a decreasing convergent sequence and that the limit γ∞ is approximately equal to .484451, give or take .0050. The integers Ak are combinatorially related to integers Ak which turn out to equal |G(k)| (theorem 10) and there is a nice recursive relation between these two sequences (theorem 11). This relationship will enable us to obtain information about the singularities of the generating functions for the sequences {Ak} and {Ak}. In addition to forming a large subset of S(g), the numerical sets in G(g) have nice properties in terms of the direct sum decompositions discussed in [AM]. Given numerical sets S and T and relatively prime atoms a ∈ A(S) and b ∈ A(T ) the direct sum of S and T is the numerical set bS ⊕ aT = {bs+ at | s ∈ S and t ∈ T}. Every numerical set S can be trivially described as S = 1S ⊕ aN for any nonzero a ∈ A(S), but if this is the only kind of direct sum decomposition of S then we say that S is irreducible. Every numerical set can be expressed as a finite direct sum of irreducibles. By [AM, Proposition 4.4], the only numerical set in ⋃ {G(g) | g ≥ 1} which is not irreducible is N1 = 2N ⊕ 3N. Thus our results show that at least 47.94% of all numerical sets in S(g) are irreducible. Another nice property is that 5We showed that |G(M)| = 1 if and only if M is symmetric, but it is not hard to construct numerical monoids M with |G(M)| = 2 that are not pseudosymmetric. 4 JEREMY MARZUOLA AND ANDY MILLER the type function is multiplicative when restricted to ⋃ {G(g) | g ≥ 1} by [AM, Proposition 5.3] (that is, the type of a direct sum is the product of the types of its factors, if the factors have no small atoms). Multiplicativity of type was a central theme in [AM]. We also mention that when a numerical set S is in G(g) its type can be computed via the formula type(S) = |S ∩ [0, g)| |S∗ ∩ [0, g)| |S ∩ S∗ ∩ [0, g)|2 , which is readily derived from, but much simpler than, the general formula for the type of an arbitrary numerical set given in [AM]. In the last two sections of the paper we explore parallel ideas for counting the number of symmetric numerical sets in G(g). This study is suggested by Backelin’s examination of the number of symmetric numerical monoids in S(g) in [B]. We show that the ratio γ σ g of the number of symmetric numerical sets in G(g) by the total number of symmetric numerical sets with Frobenius number g has a limit γ σ ∞ which is approximately equal to .23644. We also obtain information about the singularities of the generating function for the sequence {γ σ g }. In many ways the analysis of {γ σ g } turns out to be more elementary than that of {γg}. For example, in the symmetric setting we obtain two recursively related sequences of integers {A k } and {A ′ k }, and the odd terms in the second of these sequences coincides with a well-known sequence consisting of the numbers of additive 2-bases for k. Numerical sets with no small atoms Let S be a numerical set with Frobenius number g. A small atom for S is a (nonzero) atom for S which is less than g. A small atom for S is said to be large if it is greater than g/2. The first result says that every numerical set which has a small atom will have a large small atom. Lemma 3. Let S be a numerical set in S(g). If S has a small atom then S has a small atom larger than g/2. Proof. If g is even then g/2 is not an atom of S since g/2 + g/2 / ∈ S. Suppose S has an atom less than g/2 and let k be the largest such atom. Then 2k is a small atom of S, and 2k is greater than g/2 by the choice of k. The set S(g) is partitioned into two subsets G(g) = G(Ng) = {S ∈ S(g) | S has no small atoms} and B(g) = {S ∈ S(g) | S has at least one small atom} . For each g > 0, Ng ∈ G(g) and G(g) is nonempty. On the other hand, B(g) contains all of the numerical monoids in S(g) other than Ng, and B(g) is nonempty when g > 2. We are interested in the two ratios βg = |B(g)| |S(g)| = |B(g)| 2g−1 and γg = |G(g)| |S(g)| = |G(g)| 2g−1 . Observe that 0 ≤ βg, γg ≤ 1 and that βg + γg = 1. COUNTING NUMERICAL SETS WITH NO SMALL ATOMS 5 For each S ∈ S(2n− 1) and ∈ Z2 = {0, 1} we define S′ = ( S ∩ [0, n− 1] ) ∪ { n} ∪ ( 1 + S ∩ [n,∞) ) .