Duality relation for the Hilbert series of almost symmetric numerical semigroups

We derive the duality relation for the Hilbert series H (d (m) ; z) of the almost symmetric numerical semigroup S (d (m) ) combining it with its dual H (d (m) ; z (-1)). We establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F (j) , generators d (i) and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and sufficient conditions for the minimal set d (m) to generate such semigroups.