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Combinatorics of ascent sequences and their generalizations
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Resumo(s)
In 2009, Bousquet-Melou and coauthors laid the foundation of a bijective framework that relates sev- ´ eral combinatorial structures, including (2+2)-free posets and permutations avoiding a bivincular pattern of length three. To link these objects and to count them, they introduced an auxiliary class of sequences, the ascent sequence. These are defined as nonnegative integer sequences in which each entry is bounded by the number of ascents in the preceding prefix. Since then, ascent sequences have been progressively generalized to enumerate broader classes of these structures. In 2023, Benyi, Claesson and Dukes intro- ´ duced weak ascent sequences. These are defined analogously to the classical case, but (strict) ascents are replaced with weak ascents. In the spirit of the original framework, the authors provided bijections with several classes of permutation, posets and matrices. By further relaxing the growth condition on the rightmost entry, replacing ascents or weak ascents with difference-d ascents, Dukes and Sagan introduced the class of d-ascent sequences. They established a bijection between d-ascent sequences and upper-triangular matrices subject to a column restriction, and provided natural injections into other structures, such as permutations avoiding a bivincular pattern of length d +3 and factorial posets avoiding a specially labeled poset with d +3 elements. The problem of upgrading these injections to bijections was recently solved by Zang and Zhou, who introduced d-permutations and difference d posets and proved their bijective correspondence with d-ascent sequences. In this thesis we present and illustrate the bijections between these three families of ascent sequences and three combinatorical objects: permutations, matrices and posets. The main goal of our study is to understand how these maps evolve as the underlying structures are progressively generalized.
Descrição
Tese de mestrado, Matemática , 2025, Universidade de Lisboa, Faculdade de Ciências
Palavras-chave
ascent sequences weak ascent sequences difference ascent sequences Fishburn structures