Rationality, irrationality, and Wilf equivalence in generalized factor order

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 component wise 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≥uw 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 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 function M(u)=∑w≥u|μ(u,w)|w can be irrational.