Abstract

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.

Highlights

  • In what follows, N = {0, 1, 2, . . .} denotes the set of nonnegative integers

  • Theorem 2, whose proof is deferred until Section 4, 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))

  • For each key factorization-length statistic we provide an explicit, asymptotically equivalent expression when available

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Summary

Introduction

Investigations usually concern sets of lengths (i.e., without repetition), including asymptotic structure theorems [31, 37, 40, 52] as well as specialized results spanning numerous families of rings and semigroups from number theory [7,8,17], algebra [5,6] and elsewhere (see the survey [38] and the references therein). 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. Theorem 2, whose proof is deferred until Section 4, 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)) This result is of independent interest to the numerical semigroup community, its true power emerges when combined with Theorems 1 and 3.

Applications and Examples
Complete homogeneous symmetric polynomials
Proof of Theorem 2
Proof of Theorem 3
Proof of Theorem 1
Concluding remarks
Full Text
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