Rationality, irrationality, and Wilf equivalence in generalized factor order
Kitaev, Sergey and Liese, Jeff and Remmel, Jeffrey and Sagan, Bruce (2009) Rationality, irrationality, and Wilf equivalence in generalized factor order. The Electronic Journal of Combinatorics, 16 (2). R22. ISSN 10778926
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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 transfermatrix 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≥uμ(u,w)w can be irrational.


Item type: Article ID code: 49910 Dates: DateEvent2009PublishedSubjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 21 Oct 2014 10:30 Last modified: 08 Apr 2024 21:45 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/49910