Factoring in the Chicken McNugget Monoid

Abstract

We use the Chicken McNugget Monoid to demonstrate various factorization properties related to relations and chains of factorizations. We study in depth the catenary and tame degrees of this monoid.Luck is a dividend of sweat. The more you sweat, the luckier you get.-Ray Kroc [26] arXiv:1912.04494v1 [math.CO]

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

“…The publication of [3] led to a long series of papers devoted to the study of delta sets and related properties in numerical semigroups, which approach delta sets from both theoretical and computational standpoints. In our bibliography, we offer a subset of this list of papers that include undergraduate co-authors ( [1,4,5,7,8,9,16]).…”

“…Given how central the functions that compute Z(n) and L(n) are, these functions have undergone numerous improvements since the early days of the numericalsgps package, and now run surprisingly fast even for reasonably large input. McN); [ 3,7,8,9,10 ] gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(150, McN); [ 10,11,13,14,15,16,17,18,19,20,21,22,23,24,25 ] The numericalsgps package can also compute delta sets, both of numerical semigroups and of their elements. The original implementation of the latter function used Theorem 5 to compute the delta set of every element up to N, and only more recently was a more direct algorithm developed [14].…”

“…Although numerical semigroups may seem opaque, there are some very practical ways to think about them. For many years, McDonald's sold Chicken Mc-Nuggets in packs of 6, 9, and 20, and as such, it is possible to buy exactly n Chicken McNuggets using only those three pack sizes precisely when n ∈ 6, 9,20 . For this reason, the numerical semigroup S = 6, 9, 20 is known as the McNugget semigroup (see [9]).…”

“…For many years, McDonald's sold Chicken Mc-Nuggets in packs of 6, 9, and 20, and as such, it is possible to buy exactly n Chicken McNuggets using only those three pack sizes precisely when n ∈ 6, 9,20 . For this reason, the numerical semigroup S = 6, 9, 20 is known as the McNugget semigroup (see [9]). It turns out that it is impossible to buy exactly 43 Chicken McNuggets using only packs of 6, 9, and 20, but for any integer n > 43, there is some combination of packs that together contain exactly n Chicken McNuggets.…”

Elasticity in Apéry Sets

“…A numerical semigroup S is a subset of Z ≥0 with finite complement that is closed under + and contains 0. Numerical semigroups (which are, in fact, monoids, since they contain 0) have been the subject of considerable recent study [9,12,[15][16][17][18]. Many applications are known, such as in coding theory [7], algebraic geometry [6], and discrete optimization [19].…”