Complete partitions are a generalization of MacMahon’s perfect partitions; we further generalize these by defining k-step partitions. A matrix equation shows an unexpected connection between k-step partitions and distinct part partitions. We provide two proofs of the corresponding theorem, one using generating functions and one combinatorial. The algebraic proof relies on a generalization of a conjecture made by Paul Hanna in 2012.
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J. L. Brown, Note on complete sequences of integers. Amer. Math. Monthly 68 (1961) 557–560.
V. E. Hoggatt and C. H. King, Problem E1424. Amer. Math. Monthly 67 (1960) 593.
P. A. MacMahon, Combinatory Analysis, vol. 1. Cambridge University Press, Cambridge, 1915.
S. K. Park, Complete partitions. Fibonacci Quart. 36 (1998) 354–360.
S. K. Park, The \(r\)-complete partitions. Disc. Math. 183 (1998) 293–297.
R. Schneider, Arithmetic of partitions and the \(q\)-bracket operator. Proc. Amer. Math. Soc. 145 (2017) 1953–1968.
N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences. Published electronically at oeis.org, 2019.
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Andrews, G.E., Beck, G. & Hopkins, B. On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions. Ann. Comb. (2020). https://doi.org/10.1007/s00026-019-00476-1
- Integer partitions
- Distinct part partitions
- Complete partitions
Mathematics Subject Classification