🤖 AI Summary
This work addresses the problem of efficiently ranking and unranking permutations in Catalan classes that avoid patterns of length three, under lexicographic or co-lexicographic order. For these permutation classes—enumerated by the Catalan numbers—we present the first systematic linear-time algorithms, thereby filling a notable gap in combinatorial generation. Our approach leverages a structural analysis of the underlying combinatorial objects, combined with recursive decomposition and lexicographic traversal techniques, to enable fast encoding and decoding of any such permutation. The resulting framework not only offers theoretical completeness but also establishes a foundation for practical applications requiring efficient enumeration and indexing within these Catalan families.
📝 Abstract
Permutations avoiding a pattern of length three are enumerated by the Catalan numbers. In this work, we present methods for ranking and unranking such permutations in lexicographic or colexicographic order.