Introduction

A numerical set is a cofinite subset of the nonnegative integers N 0 containing 0. A numerical set closed under addition is called a numerical semigroup. The maximum integer missing from a numerical set or semigroup is called its Frobenius number. The number of positive integers that a numerical set or semigroup does not contain is called its genus. Numerical semigroups have been the subject of considerable study (e.g. [2,4]); for a general reference see [1] or [6].

Let T be a numerical set. Set A(T ) = {n ∈ N 0 : n + T ⊆ T }. This is known to be a numerical semigroup, called its atom monoid, with A(T ) ⊆ T . For a fixed numerical semigroup S, we write N (S) to denote the number of numerical sets T satisfying A(T ) = S. Numerical sets and their atom monoids have been of interest lately due to their connection with core partitions (see [3]).

Fairly recently [5] appeared, which fixed the Frobenius number f and considered all 2 f -1 numerical sets with that Frobenius number. It focused on the numerical semigroup with Frobenius number f and maximal genus, i.e. S f = {0, f +1, f +2, . . .} = {0, f +1, →}.

It determined bounds on N (S f ), and also found the asymptotic limit lim f →∞ N (S f ) 2 f -1 to be approximately 0.48.

We wish to extend this work with Frobenius number f , from the maximum genus of f to the almost-maximum genus of f -1. Hence, we consider the semigroups S f (l) = {0, f -l, f + 1, →}. We call a numerical set T with A(T ) = S f (l) both (f, l)-good and fgood. We set N (S f (l)) to denote the number of (f, l)-good numerical sets, and N (S f ( )) to denote the number of f -good numerical sets (over all l). We now look for bounds for N (S f (l)) and N (S f ( )), as well as the asymptotic limit lim f →∞

, as this is a semigroup and hence closed under addition; this will render the result no longer of the desired genus. Hence we must have l < f 2 , and thus

)). For a numerical set T and x ∈ T , we say that y is a witness to x if y ∈ T and x+y / ∈ T . This leads to a simple characterization of A(T ), for all numerical sets. Proposition 1.1 Given numerical set T and x ∈ T , x / ∈ A(T ) if and only if there is some witness to x.

Proof. If y is a witness to x, then x + y ∈ x + T but x + y / ∈ T , so x / ∈ A(T ). If there is no witness to x, then for all y ∈ Z, if y ∈ T then x + y ∈ T ; hence x + T ⊆ T and thus x ∈ A(T ).

Suppose that T is an (f, l)-good numerical set. For x = f -l and for x > f , we must have x ∈ T since A(T ) ⊆ T . Also, f / ∈ T since T, A(T ) share the same Frobenius number. We now present a result specific to our S f (l) context. Proposition 1.2 Let T be an (f, l)-good numerical set, and

∈ T , then x would be a witness to f -l, and hence f -l / ∈ T . But this is impossible since T is (f, l)-good.

Upper Bounds

In this section we provide some structural information about (f, l)-good sets, as well as an upper bound for their number.

Recall that if T is an (f, l)-good numerical set, then f -l ∈ T . Hence l / ∈ T , or else by Proposition 1.2 we would have l

a union of three intervals of length l -1, f -2l -1, and l -1, respectively. All (f, l)-good numerical sets consist of a subset of Y , together with all of S f (l). Hence, naively we get an upper bound for N (S f (l)) of 2 |Y | = 2 f -3 . We use Proposition 1.2 to improve this. Theorem 2.1 For fixed l, f , the number of (f, l)-good numerical sets N (S f (l)) satisfies

the pump journal of undergraduate research 3 (2020), 62-67 Proof. For each x ∈ {1, 2, . . . , l -1}, we have x + f -l ∈ {f -l + 1, . . . , f -1}. This yields l -1 pairs {x, x + f -l}. By Proposition 1.2, if T is (f, l)-good and x ∈ T , then x + f -l ∈ T . Hence each pair gives three possibilities: neither element in T , both elements in T , or just x + f -l ∈ T . The fourth possibility, of just x ∈ T , is forbidden. This reduces the naive upper bound by a factor of (3/4) l-foot_0 . Corollary 2.2 For a fixed f , the number of f -good numerical sets N (S f ( )) satisfies

Proof. Set t = (f -1)/2 , and we have

Corollary 2.2 bounds N (S f ( )) away from its maximum value of 2 f -1 , proving that not all numerical sets are good 1 . Unfortunately, it is not sufficient to bound the asymptotic limit lim f →∞

away from 1, much less away from 0.52.

Lower Bounds

We now turn to a lower bound for N (S f (l)), which we provide in the following.

Theorem 3.1 For fixed l, f , the number of (f, l)-good numerical sets N (S f (l)) satisfies

Proof. We will define l-1

subsets of Y , each of which may independently be included, or not, in an (f, l)-good numerical set.

First, for x ∈ {1, 2, . . . , l-1 2 }, we consider the set

. Also, note that

leaving the subset {l + 1, . . . , f -l -1} of Y undisturbed. Note that for each y ∈ Q x , also f -y ∈ Q x , and these are witnesses for each other as their sum is f / ∈ T . Hence, if

For each y ∈ R x , also f -y ∈ R x . These are witnesses for each other, and so if

Let T contain S f (l), together with an arbitrary collection of the subsets Q x , R x . In particular, l, f / ∈ T and f -l ∈ T . By the above, Y ∩ A(T ) = ∅. It is easy to see that 0 ∈ A(T ), f / ∈ A(T ), and x ∈ A(T ) for all x > f . The only remaining concern is to prove that f -l ∈ A(T ). Suppose instead that f -l / ∈ A(T ). Then there would be some witness y ∈ T with y + f -l / ∈ T . Note that if y ≥ l + 1, then y + f -l ≥ f + 1, and so y + f -1 ∈ T and y cannot be a witness. In particular, it could not be among the R x sets. If there is some x with y ∈ Q x , then either y = x or y = l -x (else y ≥ l + 1 again). But for both of these choices, y + f -l ∈ Q x again, so y is again not a witness. Hence f -l ∈ A(T ). Corollary 3.2 For a fixed f , the number of f -good numerical sets N (S f ( )) satisfies

Proof. We begin with l-1

2 . Set t = (f -1)/2 , and we have

The sum is a geometric series, and thus

Although Corollary 3.2 provides a nontrivial lower bound for N (S f ( )), it is not sufficient to bound the asymptotic limit lim f →∞ N (S f ( )) 2 f -1 away from 0. We conjecture that this holds, and, more strongly, that for a fixed l, lim f →∞ N (l,f ) 2 f -1 ∈ (0, 1). We lastly observe that preprint [7] has very recently been made public, extending the above work, addressing our conjectures, and bounding the asymptotic limit lim f →∞ N (S f ( )) 2 f -1 away from 0.

the pump journal of undergraduate research 3 (2020), 62-67