On the deviation and the type of certain local Cohen-Macaulay rings and numerical semigroups

In J. Herzog and E. Kunz (1973) [6] it was shown that for any pair (d,t)∈N×N+ with (d,t)≠(1,1) there exists a local Cohen-Macaulay ring R having deviation d(R)=d and type t(R)=t. By E. Kunz (1974) [7] the case d(R)=1,t(R)=1 cannot occur. In this paper certain Cohen-Macaulay rings are studied for which there are close relations between deviation, type and embedding dimension. Similar relations for other classes of local rings have been proved in the recent paper by L. Sharifan (2014) [15]. Our relations will be applied to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing E. Kunz (2016) [8] with ring-theoretic proofs, in which a question of H. Wilf (1978) [16] has a positive answer.