Transition matrices and Pieri-type rules for polysymmetric functions (accepted in Algebraic Combinatorics)
Asvin G and Andrew O'Desky introduced the idea of polysymmetric functions in their paper. These can be understood as a tensor product of symmetric functions where the degrees in the ith factor are scaled by i. One family of bases for the algebra of polysymmetric functions can be constructed by forming tensor products of symmetric functions. In their paper, they also introduce non-pure bases which they name H, E+, E and P. We provide combinatorial interpretations for the coefficients of transition matrices between F = H, E+, E, P and the tensor bases formed by m, s, p.
A local framework for proving combinatorial matrix inversion theorems (2025)
Many objects in combinatorics can be constructing by repeated addition of incremental structures. Some examples of such constructions are Pieri rules, Murnaghan-Nakayama rules etc. Let A and B be (rectangular) matrices where the entries count (signed, weighted) objects constructed via incremental structures. In this paper, we provide combinatorial interpretations of AB = I as local manipulations of these incremental structures. We prove this for many families of matrices and in some cases, provide canonical bijections between the pairs of objects that appear in the product AB.
Enumeration of Partitions modulo 4 (v1 2022, v2 2023)
We study the modular properties of the degrees of the irreducible representations of symmetric groups as well as alternating groups. The modulo 2 behavior of the irreducible representations of symmetric groups is well-known, due to Macdonald and McKay. We extend these results modulo 4 for symmetric/alternating groups on n letters for some natural numbers n.
Counting Cores and Bar cores: From Modular Forms to McKay numbers (You can request a copy here or email me!)
In this thesis, we delve into the theory of cores of partitions and tackle two main problems: the enumeration of t-cores and t-bar cores, and the computation of McKay numbers for the symmetric and alternating groups. For the enumeration problem, we discuss new and known explicit results for small values of t and bounds for general values of t which are obtained through the theory of modular forms. We also present new generating functions for t-bar cores in the case when t is even. For the computation of McKay numbers, we invoke the theory of p-core towers, for primes p, which serves as a direct application of the topic of enumeration of p-cores to other combinatorial problems. We also resolve the values for p=2 further and study them modulo 4. This thesis presents itself as a survey of existing results in literature that have paved the way towards solving the two problems and as a presentation of original contributions in the same direction.