For numerical semigroups, extremal p-lengths are eventually quasipolynomial with explicit degree, period, and leading coefficient; for arithmetical congruence monoids, p-lengths of x^n grow like Θ(n), Θ(n^(1/2)), or Θ(n^(2/3)) depending on the element.
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Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids
For numerical semigroups, extremal p-lengths are eventually quasipolynomial with explicit degree, period, and leading coefficient; for arithmetical congruence monoids, p-lengths of x^n grow like Θ(n), Θ(n^(1/2)), or Θ(n^(2/3)) depending on the element.