On Finite Analogs of Schmidt's Problem and Its Variants

2022/05/01 by Alexander Berkovich, Berkovich, Alexander, Ali Kemal Uncu +1
Mathematics · #05A15 #05A17 #05A19 #11B34 #11B75 #11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A15 #msc:05A17 #msc:05A19 #msc:11B34 #msc:11B75 #msc:11P81

paper · pdf · doi:10.48550/arxiv.2205.00527

13 pages, 2 figures, 10 tables

arxiv created 2022/05/19 · arxiv updated 2022/05/20

Abstract

We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szegő polynomials by Berkovich-Warnaar, and present various Schmidt's problem alike theorems and their refinements. 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.

Related

• Schmidt-type theorems for partitions with uncounted parts
• Schmidt Type Partitions
• Systematic study of Schmidt-type partitions via weighted words
• An Overpartition Companion of Andrews and Keith's 2-colored q-series Identity
• New Weighted Rogers-Ramanujan Partition Theorems and their Implications