Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions.


Introduction
In what follows, N = {0, 1, 2, . . .} denotes the set of nonnegative integers. A numerical semigroup S ⊂ N is an additive subsemigroup containing 0. We write S = n 1 , n 2 , . . . , n k = {a 1 n 1 + a 2 n 2 + · · · + a k n k : a 1 , a 2 , . . . , a k ∈ N} for the numerical semigroup generated by distinct 1 ≤ n 1 < n 2 < · · · < n k in N. Each numerical semigroup S admits a finite generating set. Moreover, there is a unique generating set that is minimal with respect to containment [42]. We always assume S has finite complement in N or, equivalently, that gcd(n 1 , n 2 , . . . , n k ) = 1. However, we do not assume that n 1 , n 2 , . . . , n k minimally generates S.
A factorization of n ∈ S is an expression n = a 1 n 1 + a 2 n 2 + · · · + a k n k of n as a sum of generators of S, which we represent here with the k-tuple a = (a 1 , a 2 , . . . , a k ) ∈ N k . The length of the factorization a is a = a 1 + a 2 + · · · + a k .
The length multiset of n, denoted L n , is the multiset with a copy of a for each factorization a of n. Recall that a multiset is a set in which repetition is taken into account; that is, its elements can occur multiple times. In particular, the cardinality |L n | of L n equals the number of factorizations of n.
It is well known that all sufficiently large n ∈ N belong to S when the generators are relatively prime. The largest integer that does not belong to S, called its Frobenius number, has been studied extensively in the literature [41]. More recently, Bourgain and Sinai [13], among others [2,18], investigated the asymptotic behavior of the Frobenius number, as did Arnold [4] in the context of estimating the number of factorizations of elements of S.
Factorizations and their lengths have been studied extensively under the broad umbrella of factorization theory (see [28] for a thorough introduction). Investigations usually concern sets of lengths (i.e., without repetition), including asymptotic "structure theorems" [25,27,36] as well as specialized results spanning numerous families of rings and semigroups from number theory [7,8,14], algebra [5,6] and elsewhere (see the survey [26] and the references therein). Several combinatorially-flavored invariants have also been studied (e.g., elasticity [3] and the delta set [30]) to obtain more refined comparisons of length sets across different settings. Numerical semigroups have received particular attention [29,38], in part due to their suitability for computation [20,24] and the availability of machinery from combinatorial commutative algebra [33,35,37]. Additionally, factorizations of numerical semigroup elements arise naturally in optimization as solutions to knapsack problems [19,40] as well as in algebraic geometry and commutative algebra [1,10].
Surprisingly absent from the literature, in all of the aforementioned settings, are results concerning the multiplicities of factorization lengths. We consider here asymptotic questions surrounding length multisets, beginning with numerical semigroups. This question was initially studied in [23] for three-generated numerical semigroups, where a closed form for the limiting distribution was obtained via careful combinatorial arguments for bounding factorization-length multiplicities. This approach proved difficult, if not impossible, when four or more generators are allowed and [23] ended with many questions unanswered.
Theorem 1 below, our main result, answers almost all questions about the asymptotic properties of important statistical quantities associated to factorization lengths in numerical semigroups. It relates asymptotic questions about factorization lengths to properties of an explicit probability distribution, which permits us to obtain numerous asymptotic predictions in closed form. Our theorem recovers the key results from [23] on three-generated semigroups, and generalizes them to semigroups with an arbitrary number of generators, using both measure-theoretic techniques and an algebraic combinatorial identity.
For what follows, we require some algebraic terminology. The complete homogeneous symmetric polynomial of degree p in the k variables x 1 , x 2 , . . . , x k is in which the coefficients c 1 (n), c 2 (n), . . . , c d (n) are periodic functions of n ∈ Z [11]. A quasirational function is a quotient of two quasipolynomials. The cardinality of a set X is denoted |X|. Theorem 1. Let S = n 1 , n 2 , . . . , n k with gcd(n 1 , n 2 , . . . , n k ) = 1 and k ≥ 3.
(a) For real α < β, where F : R → R is the probability density function (c) For any continuous function g : (0, 1) → C, In Theorem 1a, observe that F is a piecewise-polynomial function of degree k − 2 that is (k − 3)-times continuously differentiable, but not everywhere differentiable k − 2 times. In particular, its smoothness increases as the number of generators increases.
The explicit nature and broad generality of Theorem 1 permit strikingly accurate asymptotic predictions, often in closed form, of virtually every statistical quantity related to factorization lengths. For example, Theorem 1 immediately predicts the number of factorizations of n, the moments of the factorizationlength multiset L n , its mean, standard deviation, median, mode, skewness, and so forth (see Section 2). The flexibility afforded by Theorem 1c permits us to address quantities such as the harmonic and geometric mean factorization length, which would previously have been beyond the scope of standard semigrouptheoretic techniques.
The proof of Theorem 1 is contained in Section 5. It involves a variety of tools that are not standard fare in the numerical semigroup literature. For example, weak convergence of probability measures, Fourier transforms of distributions, and the theory of characteristic functions come into play. In addition, Theorem 1 builds upon two other results, described below, whose origins are in complex variables (Theorem 2) and algebraic combinatorics (Theorem 3).
The following two results, which are both critical to the proof of Theorem 1, are also of interest in their own right. The first, whose proof is deferred until Section 3, concerns a quasipolynomial representation for the pth power sum of the factorization lengths of n (the main ingredient for the pth moment of F(x)). Although this result is of independent interest to the numerical semigroup community, its true power emerges when combined with Theorems 1 and 3. Theorem 2. Let S = n 1 , n 2 , . . . , n k with gcd(n 1 , n 2 , . . . , n k ) = 1 and p ∈ N. Then in which w p (n) is a quasipolynomial of degree at most k + p − 2 whose coefficients have period dividing lcm(n 1 , n 2 , . . . , n k ).
Our next result, whose proof is in Section 4, is an exponential generating function identity. Although its derivation involves a bit of algebraic combinatorics and the result itself might seem a bit of a digression, this identity is a crucial ingredient in Theorem 1.
We are optimistic that Theorem 1 will prove to be a standard tool in the study of numerical semigroups; statistical results about factorization lengths that before appeared intractable are now straightforward consequences of Theorem 1. We devote all of Section 2 to applications and examples of our results. Sections 3, 4, and 5 contain the proofs of Theorems 2, 3, and 1 respectively. We wrap up in Section 6 with some closing remarks.

Applications and Examples
This section consists of a host of examples and applications of Theorems 1 and 2. We avoid the traditional corollary-proof format, which would soon become overbearing, in favor of a more leisurely and less staccato pace. In particular, we demonstrate how a wide variety of factorization-length statistics, some frequently considered and others more exotic, can be examined using our methods. The following examples and commentary illustrate the effectiveness of our techniques as well as their implementation.
We begin in Subsection 2.1 with a brief rundown of fundamental factorizationlength statistics, giving closed-form formulas for the asymptotic behavior when convenient. In Subsection 2.2, we recover all of the key results of [23] on threegenerator numerical semigroups. Subsection 2.3 contains explicit formulas, all of them novel, for asymptotic statistics in four-generated semigroups. Numerical semigroups with more generators and related phenomena and are discussed in Subsection 2.4.
2.1. Factorization-length statistics. Fix S = n 1 , n 2 , . . . , n k , where as always we assume that gcd(n 1 , n 2 , . . . , n k ) = 1. The quasipolynomial or quasirational functions mentioned below all have Q-valued coefficients with periods dividing lcm(n 1 , n 2 , . . . , n k ). For each key factorization-length statistic we provide an explicit, asymptotically equivalent expression when available. We say that f (n) ∼ g(n) if lim n→∞ f (n)/g(n) = 1 and f (n) = O(g(n)) if there is a constant C such that | f (n)| ≤ C|g(n)| for sufficiently large n ∈ N.
(a) Number of Factorizations. Theorem 2 with p = 0 implies that the cardinality |L n | of the factorization length multiset L n is a quasipolynomial and (b) Moments. Theorem 2 and (4) imply that the pth factorization length moment is quasirational.
(c) Mean. The preceding implies that the mean factorization length is quasirational. It is asymptotically linear as n → ∞ and its slope is the reciprocal of the harmonic mean of the generators of S.
(d) Variance and standard deviation. The factorization length variance, given by σ 2 (n) := m 2 (n) − (m 1 (n)) 2 , is quasirational by (b). From (5), we have The standard deviation is then σ(n), the square root of the variance. (g) Skewness. The factorization length skewness is the third centered moment. In light of (b), (c), and (d), an explicit asymptotic formula for Skew L n can be given, although we refrain from doing so.  The asymptotic length distribution function F(x) for a three-generated semigroup S = n 1 , n 2 , n 3 is a triangular distribution on [1/n 3 , 1/n 1 ] with peak of height 2n 1 n 3 /(n 3 − n 1 ) at 1/n 2 .
(i) Harmonic mean. The harmonic mean factorization length satisfies The integral is taken over [0, 1] for convenience; since F is supported on [1/n k , 1/n 1 ], the integrand vanishes at t = 0.
(j) Geometric mean. The geometric mean factorization length satisfies For the sake of uniformity, we often prefer to use the more explicit notation Mean L n , Median L n , Mode L n , Var L n , StDev L n , HarMean L n , Skew L n , and GeoMean L n , instead of distinctive symbols, such as µ(n) or σ(n).

Three generators: triangular distribution.
The asymptotic behavior of factorization lengths in three-generator semigroups was studied in [23] with other methods. Theorem 1 recovers all of the main results from that paper.
For S = n 1 , n 2 , n 3 , the function F(x) of Theorem 1 is a triangular distribution; see Figure 1. Indeed, letting k = 3 and (a, b, c) = ( 1 n 3 , 1 This is the familiar triangular distribution on [a, b] with peak of As predicted in the comments after Theorem 1, the distribution function is continuous but not everywhere differentiable. The standard properties of the triangular distribution provide us with the asymptotic behavior of lengths in three-generated semigroups: Var L n ∼ n 2 18 , and The harmonic and geometric means can also be worked out in closed form; the interested reader may wish to pursue the matter further.

Example 7.
Consider the McNugget semigroup S = 6, 9, 20 . The normalized histogram of the length multiset L n rapidly approaches the corresponding triangular distribution with parameters (a, b, c) = ( 1 20 , 1 9 , 1 6 ); see Figure 2. The asymptotic formulae furnished by our results perform admirably in estimating key factorization-length statistics; see Table 1.   Table 1. Actual versus predicted statistics (rounded to two decimal places) for L 10 5 , the multiset of factorization lengths of 100,000, in S = 6, 9, 20 .
2.3. Four generators: piecewise quadratic. For S = n 1 , n 2 , n 3 , n 4 , the function F(x) of Theorem 1 is piecewise quadratic and can be worked out in closed form: We remark that this explicit formula for the length distribution function completely answers the open problem suggested at the end of [23]. As predicted by the comments after Theorem 1, F is continuously differentiable but not twice differentiable. Moreover, one can see that F is unimodal and that its absolute maximum is attained in ( 1 n 3 , 1 n 2 ). A few computations reveal that  Mode L n ∼ n 1 n 2 − n 3 n 4 n 1 n 2 n 3 + n 1 n 2 n 4 − n 1 n 3 n 4 − n 2 n 3 n 4 n, and The asymptotic median factorization length is not so amenable to closed-form expression, although it is easily computed for specific semigroups as we see below.
Example 8. For S = 11, 34, 35, 36 , we have see Figure 3. Elementary computation confirms that the median of the distribution function F(x) occurs in [ 1 34 , 1 11 ]. For x ∈ [ 1 34 , 1 11 ], we find that attains the value 1 2 at precisely one point, namely 1 11 (1 − 3 115 714 ) ≈ 0.041. Thus,   Example 9. The factorization-length skewness, being expressible in terms of the first and third moments, and the variance, can be given in closed form: in which (a, b, c, d) = ( 1 n 1 , 1 n 2 , 1 n 3 , 1 n 4 ). In particular, Skew L n tends to zero (that is, F tends to be highly symmetric) if and only if one of the following occurs: For example, the equalities yield two numerical semigroups with highly symmetric length distribution functions; see Figure 4. This highlights another connection between Egyptian fractions and the statistical properties of length distributions in numerical semigroups [23].
Similar computations can be carried out for semigroups with more generators, although it becomes rapidly less rewarding to search for answers in closed form as the number of generators increases. We leave the details and particulars of such symbolic computations to the reader.

Additional examples.
In this section, we give two final examples. The first points out a curious, but easily explained, phenomenon related to the constant δ = gcd(n k − n k−1 , n k−1 − n k−2 , . . . , n 2 − n 1 ), which arises in the semigroup literature as the minimum element of the delta set (see [17] for more on this invariant). Since n i ≡ n j (mod δ) for every i, j, it follows that δ is the smallest distance that can occur between distinct factorization lengths of n, meaning all factorization lengths of a given n are equivalent modulo δ. If δ > 1, then this causes "gaps" between positive values in the length multiset.
Example 10. Let S = 9, 11, 13, 15, 17 , for which δ = 2. If n is even, then every element of L n is even, and if n is odd, then every element of L n is odd. The corresponding length distribution function is x, which appears to be half the height of the upper curve suggested by the blue dots in Figure 5 since the factorization lengths of n all have identical parity. In other words, the upper curve suggested by the blue dots in Figure 5 must be "averaged out" by δ to produce the red curve, which depicts F(x). The predictions afforded by our methods in this case, as outlined in Table 3, are still surprisingly accurate.
We conclude with one final example that demonstrates the impressive estimates our techniques afford for a numerical semigroup with nine generators.  Table 3. Actual versus predicted statistics (rounded to two decimal places) for L 10 5 , the multiset of factorization lengths of 100,000, in S = 9, 11, 13, 15, 17 .  is far beyond the realm of previously-established techniques. We spare the reader the typographically-unwieldy display of the explicit length-generating function; suffice it to say, F is a piecewise polynomial function of degree 7 (see Figure 6). A few computations with a computer algebra system provide accurate approximations to the relevant statistics; see Table 4.

Proof of Theorem 2
The first part of the proof concerns a certain two-variable generating function (Subsection 3.1). Next comes a lengthy residue computation (Subsection 3.2). A few power series computations complete the proof (Subsection 3.3).

Lemma 14.
For p ∈ N, Proof. We proceed by induction. The base case p = 0 is (12). For the inductive step, suppose that (15) holds for some p ∈ N. Then The final equality follows from counting how many times each term appears when the line before it is expanded.
Each 1 − z n i factors as a product of n i distinct linear factors, one of which is 1 − z.
Consequently, 1 is a pole of G(z) of order k + p; this arises from the summand corresponding to i = p. Moreover, k + p is the maximum possible order for a pole of G(z), and 1 is the unique pole of this order. Indeed, gcd(n 1 , n 2 , . . . , n k ) = 1 ensures that the only common root of 1 − z n 1 , 1 − z n 2 , . . . , 1 − z n k is 1. Thus, Λ p (n) is a complex linear combination of terms of the form n r−1 ω n , n r−2 ω n , . . . , ω n , in which ω is a pole of G(z) of order at most r (see [11,Ch. 1] for an overview of this method). The unique pole of G(z) of highest order is 1, which has order k + p. Thus, there exist periodic functions a 0 , a 1 , . . . , a k+p−1 : N → C with periods dividing L such that Λ p (n) = a k+p−1 (n)n k+p−1 + a k+p−2 (n)n k+p−2 + · · · + a 1 (n)n + a 0 (n).

A residue computation.
Since G(z) has a pole of order k + p at 1, we have for some constant C and some rational function u, all of whose poles are Lth roots of unity with order at most k + p − 1. In particular, for some quasipolynomial w(n) of degree at most k + p − 2 and period dividing L. The only summand in (17) that has a pole at 1 of order k + p is the term that corresponds to i = p. The summands in (17) , · · · , z n k 1 − z n k = p! lim z→1 k ∏ j=1 1 1 + · · · + z n j −1 h p z n 1 1 + · · · + z n 1 −1 , · · · , z n k 1 + · · · + z n k −1 = p! n 1 n 2 · · · n k h p 1 n 1 , 1 n 2 , . . . , 1 n k .
3.3. Completing the proof. Observe that in which v(n) is a quasipolynomial of degree k + p − 2 with integer coefficients. Together with (18) and (19), we obtain Thus, in which q(n) is a quasipolynomial of degree at most k + p − 2 whose coefficients have periods dividing L. Additionally, since v(n) and w(n) both have rational coefficients, so must q(n). This completes the proof.

Proof of Theorem 3
We wish to prove the exponential generating function identity valid for z ∈ C. We first show that the power series on the left-hand side of (20) has an infinite radius of convergence (Subsection 4.1). Then we reduce (20) to an identity that links complete homogeneous symmetric polynomials to the determinants of certain Vandermonde-like matrices (Subsection 4.2). A brief excursion into algebraic combinatorics (Subsection 4.3) finishes off the proof.

Radius of convergence.
Fix distinct x 1 , x 2 , . . . , x k ∈ C\{0}. We claim that the radius of convergence of the power series is infinite. This ensures that (20) is an equality of entire functions. The ordinary generating function for the complete homogeneous symmetric polynomials is see [45]. The radius of convergence of the preceding power series is the distance from 0 to the closest pole 1/ a second appeal to the Cauchy-Hadamard formula tells us that the radius of convergence R of (21) satisfies Thus, the radius of convergence of (21) is infinite.

A Vandermonde-like determinant.
The determinant of the k × k Vandermonde matrix see [31, p. 37]. In what follows, V(x 1 , . . . , x r , . . . , x k ) denotes the (k − 1) × (k − 1) Vandermonde matrix obtained from V(x 1 , x 2 , . . . , x k ) by removing x r (do not confuse the carat with the Fourier transform). Cofactor expansion and the linearity of the determinant in the final column of a matrix reveals that We reindexed the final sum to reflect the fact that the matrices in the second-tolast line have repeated columns for i = 0, 1, . . . , k − 2 and hence have vanishing determinant. To establish (20), and hence Theorem 3, it suffices to show that 4.3. Some algebraic combinatorics. To establish (23) requires a small amount of algebraic combinatorics. We briefly review the notation and results necessary for this purpose; the interested reader may consult [44] for complete details. Let λ := (λ 1 , λ 2 , . . . , λ k ) denote the integer partition To such a partition we associate the polynomial which is an alternating function of the variables x 1 , x 2 , . . . , x k (interchanging any two of the variables changes the sign of the determinant). As an alternating polynomial, the preceding is divisible by The Schur polynomial in the variables x 1 , x 2 , . . . , x k corresponding to the partition λ is this is Jacobi's bialternant identity (which is itself a special case of the famed Weyl character formula). On the other hand, the first Jacobi-Trudi formula says in which h i = h i (x 1 , x 2 , . . . , x k ) and the size of the matrix is determined by the requirement that the subscripts of the complete symmetric homogeneous polynomials are positive.

Proof of Theorem 1
The proof of Theorem 1 uses a few tools, such as weak convergence and Fourier transforms of measures, that are not standard in the study of numerical semigroups. Since these ideas are not required to apply Theorem 1 and are not used elsewhere in the paper, we introduce the required concepts as needed and make no attempt to state definitions and lemmas in the greatest possible generality.
We begin with the necessary background on moments of probability measures (Subsection 5.1), Fourier transforms of measures (Subsection 5.2), and characteristic functions (Subsection 5.3). We then prove a power series convergence lemma (Subsection 5.4) to set up our use of characteristic functions. We introduce a family of singular measures (Subsection 5.5) that converge weakly to the desired probability measure. This is established using the method of characteristic functions and Lévy's continuity theorem (Subsection 5.6). We wrap things up with a dose of Fourier inversion and some detailed computations (Subsection 5.7).

Measures and moments.
A Borel measure is a measure defined on the Borel σ-algebra, the σ-algebra of subsets of R generated by the open sets. Every subset of R we consider in this paper is a Borel set. Let ν be a probability measure on The moments of ν are uniformly bounded since  The equivalence of (a) and (b) follows from the Weierstrass approximation theorem: every continuous function on [0, 1] is uniformly approximable by polynomials. The weak limit of a sequence of probability measures is a probability measure and the limit measure is unique [12, p. 336-7].
This may differ in appearance from what the reader is accustomed to. Normally one integrates over R in (25), but that is unnecessary here because ν is supported on [0, 1]. We adhere to the positive sign in the exponent of the integrand in (25), which is standard in probability theory [12,Sect. 26]. Consequently, the reader should be aware of potential sign discrepancies between what follows and formulas from their favored sources. The inverse Fourier transform of a suitable f : With much additional work, the inverse Fourier transform can be defined on distributions ("generalized functions"). A friendly introduction to Fourier transforms of distributions is [39,Ch. 4]. In the sense of distributions, one can show The interchange of sum and integral is permissible because for each fixed z ∈ C the series involved converges uniformly for t ∈ [0, 1]. Since |m p (ν)| ≤ 1 for all p ∈ N, comparison with the exponential series ensures that the series above has an infinite radius of convergence and hence ϕ ν is an entire function. Moreover, for all x ∈ R since ν is a probability measure on [0, 1]. If ν 1 and ν 2 are probability measures and ϕ ν 1 = ϕ ν 2 , then ν 1 = ν 2 [12,Thm. 26.2]. Under certain circumstances, we can recover a probability measure from its characteristic function [12, p.347-8].
The following theorem of Lévy relates weak limits of probability measures to pointwise convergence of the corresponding characteristic functions [12,Thm. 26.3].

A power series lemma.
We ultimately plan to consider a sequence of probability measures ν n to which we will apply Lemma 29. To show that the associated sequence of characteristic functions converges we need the following lemma.

Lemma 30.
Suppose that |a p (n)| ≤ 1 for all n, p ∈ N. If lim n→∞ a p (n) = a p for each p, then ∑ ∞ p=0 a p (n) z p p! converges locally uniformly on C to ∑ ∞ p=0 a p z p p! Proof. Fix R > 0. Let N ∈ N be so large that Let M ∈ N be such that n ≥ M =⇒ |a p (n) − a p | R p p! < 2N for 0 ≤ p ≤ N − 1.
For k ≥ 3, induction and the definition of the derivative confirms that |x|x k−3 is k − 3 times continuously differentiable on R, but is not differentiable k − 2 times at x = 0. Since the n 1 , n 2 , . . . , n k are distinct and because the zeros of 1 − n r x belong to [0, 1], we conclude that F is k − 3 times continuously differentiable on   for any continuous function g : (0, 1) → C (since F is supported on [1/n k , 1/n 1 ] ⊂ (0, 1), the values of g outside of this interval are irrelevant). This concludes the proof of Theorem 1.

Concluding remarks
Theorem 1 (which depends upon Theorems 2 and 3) appears to answer all questions about the asymptotic behavior of factorization lengths in numerical semigroups. However, there are a few issues it does not immediately address which suggest several avenues for further exploration.