  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 {\em
hard-wired} if $c$ is always interpreted by the same element $j \in [n]$. In
this paper we show:
  1. 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.
  2. 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 an non-negative integer and $A$ is an
ultimately periodic set of non-negative integers.
