Counting numerical semigroups by genus and even gaps

Abstract

We contruct a one-to-one correspondence between a subset of numerical semigroups with genus g and γ even gaps and the integer points of a rational polytope. In particular, we give an overview to apply this correspondence to try to decide if the sequence (n g ) is increasing, where n g denotes the number of numerical semigroups with genus g.

Quasi-Ordinarization Transform of a Numerical Semigroup

“…Define, as in [9], the sequence f ω by f ω = ∑ Ω∈S ω #C(Ω, ω + 1). The first elements in the sequence, from f 0 to f 15 are ω 0 1 2 3 We remark that this sequence appears in [5], where Bernardini and Torres proved that the number of numerical semigroups of genus 3ω whose number of even gaps is ω is exactly f ω . It corresponds to the entry A210581 of The On-Line Encyclopedia of Integer Sequences [23].…”

“…The number of numerical semigroups of genus g is denoted n g . It was conjectured in [5] that the sequence n g asymptotically behaves as the Fibonacci numbers. In particular, it was conjectured that each term in the sequence is larger than the sum of the two previous terms, that is, n g n g−1 + n g−2 for g 2, being each term more and more similar to the sum of the two previous terms as g approaches infinity, more precisely lim g→∞ n g n g−1 +n g−2 = 1 and, equivalently, lim g→∞…”

“…(a 1 + a 3 ) − (a 1 + a 2 ) = β (a 1 + a 4 ) − (a 1 + a 3 ) = β. Now, A + A must contain all the elements in ( 5) and ( 6), as well as the element 2a 1 , which is not in (5), nor in (6). Since #(A + A) = 8, this means that there must be exactly one element that is both in (5) and in (6).…”

“…In particular, it was conjectured that each term in the sequence is larger than the sum of the two previous terms, that is, n g n g−1 + n g−2 for g 2, with each term being increasingly similar to the sum of the two previous terms as g approaches infinity, more precisely lim g→∞ n g n g−1 +n g−2 = 1 and, equivalently, lim g→∞ n g n g−1 = φ = 1+ √ 5 2 . A number of papers deal with the sequence Symmetry 2021, 13, 1084. https://doi.org/10.3390/sym13061084 https://www.mdpi.com/journal/symmetry n g [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Alex Zhai proved the asymptotic Fibonacci-like behavior of n g [21].…”

Quasi-ordinarization transform of a numerical semigroup

“…. Many other papers deal with the sequence n g [19,20,6,10,9,13,26,3,17,2,7,22,1,16,11,18] and Alex Zhai gave a proof for the asymptotic Fibonacci-like behavior of n g [25]. However, it has still not been proved that n g is increasing.…”