Research and projects
Research and projects
My research papers/preprints and projects can be found below.
Counting the number of τ-exceptional sequences over Nakayama algebras published in Algebras and Representation Theory. D Msapato
Abstract: The notion of a τ-exceptional sequence was introduced by Buan and Marsh in 2018 as a generalisation of an exceptional sequence for finite dimensional algebras. We calculate the number of complete τ-exceptional sequences over certain classes of Nakayama algebras. In some cases, we obtain closed formulas which also count other well known combinatorial objects and exceptional sequences of path algebras of Dynkin quivers
https://doi.org/10.1007/s10468-021-10060-y
Modular Fuss-Catalan numbers published in Discrete Mathematics, and awarded "Editors' Choice Award 2022" as one of the outstanding papers published in Discrete Mathematics. D Msapato
Abstract: The modular Catalan numbers C_{k,n}, introduced by Heing and Huang in 2016 count equivalence classes of parenthesizations of x_0 * x_1 * ... *x_n where * is a binary k-associative operation and $k$ is a positive integer. The classical notion of associativity is just 1-associativity, in which case C_{1,n} = 1 and the size of the unique class is given by the Catalan number C_n. In this paper we introduce modular Fuss-Catalan numbers C_{k,n}^{m} which count equivalence classes of parenthesizations of x_0 * x_1 * ... *x_n where * is an m-ary k-associative operation for m \geq 2. Our main results are a closed formula for C_{k,n}^{m} and a characterisation of k-associativity.
https://doi.org/10.1016/j.disc.2021.112704
The Karoubi envelope and weak idempotent completion of an extriangulated category published in Applied Categorical Structures. D Msapato
Abstract: We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.
https://doi.org/10.1007/s10485-021-09664-8
Preprints
Abstract: Suppose (C, E, s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C, E, s) to (resp. weakly) idempotent complete n-exangulated categories. We note that our methods of proof differ substantially from the extriangulated and (n + 2)-angulated cases. However, our constructions recover the known structures in the established cases up to n- exangulated isomorphism of n-exangulated categories.
https://doi.org/10.1007/s10485-023-09758-5
Abstract: We give a characterisation of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object X in the extriangulated category, the morphism 0 --> X is a deflation, and the morphism X --> 0 is an inflation.
Important Note: This result is was already known before the writing of this preprint, see Corollary 7.6 of Extriangulated Categories, Hovey Twin Cotorsion Pairs, Model Structure by Hiroyuki Nakaoka and Yann Palu. The work of this preprint was done without knowledge of this result, and is an alternative approach to the same result.
My programming projects can be found on my github at https://github.com/DixyMsapato