Abstract
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.
Highlights
Introduction and notationCatalan words are particular growth-restricted words and they represent still another combinatorial class counted by the Catalan numbers, see for instance [12, exercise 6.19.u, p. 222]
This paper contributes to a recent line of research on classical pattern avoidance on words subject to some growth restrictions by investigating connections between sequences on the On-line Encyclopedia of Integer Sequences [11] and Catalan words avoiding two patterns of length 3
The pairs of avoided patterns for which ascent sequences and Catalan words coincide, and for which the enumeration has already been considered in [2] are highlighted in the summarizing Table 2
Summary
Catalan words are particular growth-restricted words and they represent still another combinatorial class counted by the Catalan numbers, see for instance [12, exercise 6.19.u, p. 222]. We denote by Cn(π) the set of length n Catalan words avoiding π, and cn(π) = |Cn(π)| is the cardinality of Cn(π) and C(π) = ∪n≥0Cn(π). ) denote both the set of length n Catalan words avoiding each pattern in π; and cn(π) = cn(α, β, . In light of Corollary 1 it can happen that if a pattern of the avoided pair is one of the four specified in this corollary, the resulting ascent sequences are Catalan words as well. The pairs of avoided patterns for which ascent sequences and Catalan words coincide, and for which the enumeration has already been considered in [2] are highlighted in the summarizing Table 2. In order to keep the present article self-contained we fully consider these cases, our proofs being alternative to those in [2]
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