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Author ORCID Identifier

https://orcid.org/0000-0003-4776-5900

AccessType

Campus-Only Access for One (1) Year

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2022

Month Degree Awarded

May

First Advisor

Alejandro Morales

Subject Categories

Discrete Mathematics and Combinatorics

Abstract

This dissertation is in the field of Algebraic and Enumerative Combinatorics. In the first part of the thesis, we study the generalization of Naruse hook-length formula to mobile posets. Families of posets like Young diagrams of straight shapes and d-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula (NHLF). In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine d-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets by proving a major index q-analogue. We also give an inversion index q-analogue of the Naruse formula for mobile tree posets.

In the second part, we study the behavior of Hillman-Grassl correspondence on skew semi-standard Young tableaux (SSYT). Morales, Pak, and Panova gave a q- analogue of the NHLF for semi-standard tableaux of skew shapes. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes gave their q-analogue. We study the combinatorics of the Hillman-Grassl map on the skew-shaped tableaux with intention of giving a combinatorial proof of the theorem. For a skew shape, we define a new set of semi-standard Young tableaux, called the minimal SSYT, that are equinumerous with excited diagrams via a new description of the Hillman-Grassl bijection and have a version of excited moves. We also relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape.

Lastly, we study factorizations of Singer cycles in the finitely generated general linear group, GLn(Fq), into shortest factorization of reflections. For the case of the symmetric group, Den ́es gave a double-counting proof for the number of factoriza- tions of the long cycle into n − 1 transpositions, using the inherent bijection between factorizations and labeled trees. Using character theory, Lewis, Stanton, and Reiner showed that the number of shortest factorization of a Singer cycle is (qn − 1)n−1. We give Den ́es’ double-counting type proof to the case in GLn(Fq), giving a possible q-analogue of labeled trees. We give a graphical interpretation of the elements in GLn(Fq) to mimic the bijection between the factorization in the symmetric group and the labeled trees.

DOI

https://doi.org/10.7275/28638150

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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