Mappings and symmetries of partitions (Q1823942)

\textit{F. G. Garvan} [Trans. Am. Math. Soc. 305, No.1, 47-77 (1988; Zbl 0641.10009)] defined a vector partition of an integer n to be an ordered triple of ordinary partitions \(P=[P_ 1,P_ 2,P_ 3]\), where \(P_ 1\) is a partition into unequal parts and \(P_ 2\), \(P_ 3\) are unrestricted partitions into positive parts. The existence of a ``crank'' was conjectured by the author in 1944 and proved, along with various results concerning them, by \textit{G. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., N. Ser. 18, No.2, 167-171 (1988; Zbl 0646.10008)]. If p(k,n) is the number of partitions of n with crank k and \(N_ V(k,n)\) is the sum of the ``weights'' defined by Garvan, then Andrews and Garvan proved, using generating functions, that for all k and \(n>1\), \(p(k,n)=N_ V(k,n)\). The author provides a combinatorial interpretation of this result using a direct mapping of vector partitions onto ordinary partitions. He further defines an extended version of these vector partitions where \(P_ 2\) can have any number of zero parts, a ``rank set'' of the partition, and q(k,n), the number of partitions of n whose rank sets contain k. This allows an analogous result to be proved: \(q(k,n)=N_ E(k,n)\), for all n and k, where \(N_ E(k,n)\) is the sum of weights taken over all extended vector partitions of sum n, rank k. Further, \(q(k,n)=q(-k,n+k)\) and \(q(k,n)+q(-k-1,n)=p(n)\). Finally, the author shows that \(\sum_{k}k^ 2p(k,n)=2np(n)\), \(n>1\).