Almost symmetric numerical semigroups with high type

Abstract

We establish a one-to-one correspondence between numerical semigroups of genus g and almost symmetric numerical semigroups with Frobenius number F and type F − 2g , provided that F is greater than or equal to 4g − 1 .

Numerical Semigroups of small and large type

“…Corollary 3.2.1. (Proved in [1]) Given positive integers F, β such that F > 8β − 6. There is a bijection between the set of almost symmetric numerical semigroups with Frobenius number F and type F − 2β and the number of numerical semigroups with genus β.…”

“…In [1] it was proved that for a fixed α, the function T 1 (F, F − 2α) remains constant once F > 4α + 1. This was done by finding a bijection between the set of almost symmetric numerical semigroups S with F (S) = F , t(S) = F − 2α and the set of numerical semigroups with genus g. Following a similar method we show in Theorem 3.2 and Theorem 3.4 that Theorem 1.7.…”

Almost symmetric Arf partitions

“…Several equivalent conditions are given for AS -semigroups. For details, we refer to [4,5,9,13]. In Proposition 2.1, we explicitly describe the elements of P F (S) with respect to gap numbers of S .…”