Enumerative Results on the Schr\"oder Pattern Poset
Lapo Cioni, Luca Ferrari
TL;DR
This paper studies the Schr"oder pattern poset by deriving formulas for covering relations and enumerating Schr"oder words avoiding certain patterns, advancing combinatorial understanding of Schr"oder structures.
Contribution
It provides closed-form formulas for covering relations in the Schr"oder pattern poset and enumerates Schr"oder words avoiding specific patterns.
Findings
Formulas for the number of Schr"oder words covering or covered by a given word.
Enumeration of Schr"oder words avoiding specific patterns.
Enhanced understanding of the combinatorial structure of Schr"oder words.
Abstract
The set of Schr\"oder words (Schr\"oder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schr\"oder words as lattice paths. The resulting poset is called the Schr\"oder pattern poset. We find closed formulas for the number of Schr\"oder words covering/covered by a given Schr\"oder word in terms of classical parameters of the associated Schr\"oder path. We also enumerate several classes of Schr\"oder avoiding words (with respect to the length), i.e. sets of Schr\"oder words which do not contain a given Schr\"oder word.
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