Abstract :

MacMahon introduced a higher dimensional generalisation of the partition of (positive) integers -- these are called higher-dimensional partitions. The usual partitions are line or one-dimensional partitions. Two-dimensional partitions are also called plane partitions. Explicit formulae for generating functions, due to Euler and McMahon, are known only for the numbers of line and plane partitions of integers. By considering partitions in all dimensions on the same footing, we first discuss two refinements of the counting problem -- the first one is a 1967 result of Atkin, et al.[1] which implies that, for a fixed positive integer, n, one needs (n-1) independent numbers. The second refinement[2], a 2012 result of mine, implies that one only needs [(n-1)/2] independent numbers. The first refinement appears as a binomial transform while the second refinement is a more complicated transform. We conclude with a few related conjectures that need proof as well as a combinatorial interpretation. References: 1. A.O.L. Atkin, P. Bratley, I.G. Macdonald, and J.K.S. McKay, Some computations for m-dimensional partitions, Proc. Cambridge Philos. Soc. 63 (1967) 1097—1100. 2. Suresh Govindarajan, Notes on Higher Dimensional Partitions, arXiv:1203.4419, Journal of Combinatorial Theory, Series A 120 (2013) 600–622. 3. Partitions of integers in any dimension and for all integers $leq 26$http://www.physics.iitm.ac.in/~suresh/partitions.html 4. The partitions project at IIT Madrashttp://boltzmann.wikidot.com/the-partitions-project