Enumerative combinatorics
Recurrence relations for the number of labeled structures on a finite set
Proceedings of the Symposium "Rekursive Kombinatorik" on Logic and Machines: Decision Problems and Complexity
Modular Counting and Substitution of Structures
Combinatorics, Probability and Computing
The specker-blatter theorem revisited
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Application of logic to integer sequences: a survey
WoLLIC'10 Proceedings of the 17th international conference on Logic, language, information and computation
Definability of combinatorial functions and their linear recurrence relations
Fields of logic and computation
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Let C be a class of relational structures. We denote by fC(n) the number of structures in C over the labeled set {0, ..., n - 1}. For any C definable in monadic second-order logic with unary and binary relation symbols only, E. Specker and C. Blatter showed that for every m ∈ N, the function fC satisfies a linear recurrence relation modulo m, and hence it is ultimately periodic modulo m. The case of ternary relation symbols, and more generally of arity k symbols for k≥3, was left open.In this paper we show that for every m there is a class of structures Cm, which is definable even in first-order logic with one quaternary (arity four) relation symbol, such that fCm is not ultimately periodic modulo m. This shows that the Specker-Blatter Theorem does not hold for quaternary relations, leaving only the ternary case open.