Extensions and limits of the specker-blatter theorem

The original Specker-Blatter theorem (1983) was formulated for classes of structures $\mathcal {C}$ of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set $[n]$ is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter theorem does not hold for one quaternary relation (2003).If the vocabulary allows a constant symbol c, there are n possible interpretations on $[n]$ for c. We say that a constant c is hard-wired if c is always interpreted by the same element $j \in [n]$ . In this paper we show: The Specker-Blatter theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case. The Specker-Blatter theorem does not hold already for $\mathcal {C}$ with one ternary relation definable in First Order Logic FOL. This was left open since 1983.Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers $B_{r,A}$ , restricted Stirling numbers of the second kind $S_{r,A}$ or restricted Lah-numbers $L_{r,A}$ . Here r is a non-negative integer and A is an ultimately periodic set of non-negative integers.