Cantorian Tableaux and Permanents

Abstract

This article could be called &#34;theme and variations&#34; on Cantor&#39;s celebrated diagonal argument. Given a square nxn tableau T=(a_i^j) on a finite alphabet A, let L be the set of its row-words. The permanent Perm(T) is the set of words a_{π(1)}^1 a_{π(2)}^2 ...a_{π(n)}^n, where πruns through the set of permutations of n elements. Cantorian tableaux are those for which Perm(T)\cap L=\emptyset. Let s=s(n) be the cardinality of A. We show in particular that for large n, if s(n) <(1-ε) n/log n then most of the tableaux are non-Cantorian, whereas if s(n) >(1+ε) n/log n then most of the tableaux are Cantorian. We conclude our article by the study of infinite tableaux. Consider for example the infinite tableaux whose rows are the binary expansions of the real algebraic numbers in the unit interval. We show that the permanent of this tableau contains exactly the set of binary expansions of all the transcendental numbers in the unit interval.