Séminaire Lotharingien de Combinatoire, B88b (2023), 18 pp.
On Finite Analogs of Schmidt's Problem and Its Variants
Alexander Berkovich and Ali Kemal Uncu
Abstract.
We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Andrews-Paule-Uncu theorem. We approach the problem using Boulet-Stanley weights and a formula on Rogers-Szegő polynomials by Berkovich and Warnaar and present various Schmidt's problem alike theorems and their refinements. We study many variants of Schmidt's problem. In particular, we consider partitions with a bound on the largest part and uncounted odd-indexed parts. Our new Schmidt-type results include the use of even-indexed parts' sums, alternating sum of parts, and hook lengths as well as the odd-indexed parts' sum which appears in the original Schmidt's problem. We also translate some of our Schmidt's problem alike relations to weighted partition counts with multiplicative weights in relation to Rogers-Ramanujan partitions.
Received: January 17, 2023.
Accepted: June 20, 2023.
Final Version: August 13, 2023.
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