Rationality, irrationality, and Wilf equivalence in generalized factor order

Abstract

Let P be a partially ordered set and consider the free monoid P* of all words over P. If w,w′ ∈ P* then w′ is a factor of w if there are words u, v with w = uw′v. Define generalized factor order on P* by letting u ≤ w if there is a factor w′ of w having the same length as u such that u ≤ w′, where the comparison of u and w′ is done componentwise using the partial order in P. One obtains ordinary factor order by insisting that u = w′ or, equivalently, by taking P to be an antichain. Given u ∈ P*, we prove that the language F(u) = {w : w ≥ u} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u) =∑w≥u w is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider P = ℙ, the positive integers with the usual total order, so that ℙ* is the set of compositions. In this case one obtains a weight generating function F(u; t, x) by substituting txn each time n ∈ ℙ appears in F(u). We show that this generating function is also rational by using the transfer-matrix method. Words u, v are said to be Wilf equivalent if F(u; t, x) = F(v; t, x) and we can prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on P*. It follows that one always has μ(u,w) = 0,±1. Using the Pumping Lemma we show that the generating functionM(u) =∑w≥u |μ(u,w)|w can be irrational. © 2009 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.