Refinements and generalizations of Capparelli's conjecture on partitions
Caparelli's conjecture asserts the equality of the number of \(C\)- partitions and the number of \(D\)-partitions of \(n\) where a \(C\)-partition is a partition into distinct parts not congruent to \(\pm 1\) modulo 6 and a \(D\)-partition is one in which the difference between parts is at least 4 unless two parts are both divisible by 3 or add up to a multiple of 3, in which case they may differ by as little as 2. Various proofs of this conjecture now exist. This paper proves a generalization with four additional parameters which, for \(D\)-partitions, keep track of the number of parts in each residue class modulo 3 and of \(\lceil (\ell+ 2)/3\rceil\) where \(\ell\) is the largest part. Several different proofs are given.
scientific article; zbMATH DE number 783881