On partitions with initial repetitions (Q721840)

\textit{G. E. Andrews} [Acta Math. Sin., Engl. Ser. 25, No. 9, 1437--1442 (2009; Zbl 1189.05018)] introduced partitions with initial repetitions and established a theorem showing that partitions of \(n\) with initial \(k\)-repetitions are equinumerous to partitions of \(n\) in which no part has multiplicity exceeding \(2k-1\). His proof used generating functions. Later, \textit{W. J. Keith} [Ramanujan J. 27, No. 2, 163--167 (2012; Zbl 1243.05030)] provided a bijective proof by means of \(k\)-modular diagrams. The present paper gives a simplified bijection by showing that a certain decomposition of a partition can be obtained by algebraic operations instead of the \(k\)-modular diagrams. In a second part of the paper, the authors obtain several corollaries and generalizations of Andrews' theorem [loc. cit.] including certain theorems on partitions with ``early conditions'' as defined by \textit{G. E. Andrews} [in: Advances in combinatorics. In part based on the 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26--29, 2011. Dedicated to Herbert Saul Wilf on the occasion of his 80th birthday. Berlin: Springer. 57--76 (2013; Zbl 1271.05014)]. Finally, they reinterpret several formulas from Slater's 1952 list by means of certain extensions of partitions with initial repetitions.