Enumeration schemes for words avoiding permutations

The enumeration of permutation classes has been accomplished with a variety of techniques. One wide-reaching method is that of enumeration schemes, introduced by Zeilberger and extended by Vatter. In this paper we further extend the method of enumeration schemes to words avoiding permutation patterns. The process of finding enumeration schemes is programmable and allows for the automatic enumeration of many classes of pattern-avoiding words. Background The enumeration of permutation classes has been accomplished by many beautiful techniques. One natural extension of permutation classes is pattern-avoiding words. Our concern in this paper is not attractive methods for counting individual classes, but rather developing a systematic technique for enumerating many classes of words. Four main techniques with wide success exist for the systematic enumeration of permutation classes. These are generating trees, insertion encoding, substitution decomposition, and enumeration schemes. In this paper we adapt the method of enumeration schemes, first introduced for permutations by Zeilberger and extended by Vatter to the case of enumerating pattern-restricted words. Definition 1.1. Let [ k ] n denote the set of words of length n in the alphabet {1, …, k }, and let w ∈ [ k ] n , w = w 1 … w n . The reduction of w , denoted by red(w) , is the unique word of length n obtained by replacing the i th smallest entries of w with i , for each i .