Rationality, Irrationality, and Wilf Equivalence in Generalized Factor Order

  • Sergey Kitaev
  • Jeffrey Liese
  • Jeffrey Remmel
  • Bruce E. Sagan

Abstract

Let P be a partially ordered set and consider the free monoid P of all words over P. If w,wP then w is a factor of w if there are words u,v with w=uwv. Define generalized factor order on P by letting uw if there is a factor w of w having the same length as u such that uw, 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 uP, we prove that the language F(u)={w : wu} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u)=wuw 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 nP 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)=wu|μ(u,w)|w can be irrational.

Published
2009-12-02