Generalization of Dyson's rank and non-Rogers-Ramanujan partitions
The rank of a partition is a statistic that splits the set of partitions of \(5n+ 4\) and \(7n+ 5\) into, respectively, five and seven equinumerous sets, thus proving that \(p(5n+ 4)\equiv 0\pmod 5\) and \(p(7n+ 5)\equiv 0\pmod 7\). The crank is a similar statistic that separates the partitions of \(11n+ 7\) into eleven equinumerous sets. The author describes a \(k\)- rank for which the rank is the 2-rank and the crank is the 1-rank, and defines a generalized notion of conjugation, \(k\)-conjugation, which reverses the sign of the value of the \(k\)-rank. The author proves that the number of self-\(2k\)-conjugate partitions of \(n\) is equal to the number of partitions of \(n\) in which no part is divisible by \(2k\) and parts divisible by \(k\) are distnct.
number of self-\(2k\)- conjugate partitions
scientific article; zbMATH DE number 703888
scientific article