Annals of Combinatorics 4 (2000) 327-338MacMahon's Partition Analysis: II Fundamental Theorems George E. Andrews Department of Mathematics,
The Pennsylvania State University, University Park, Pennsylvania
16802, USA Received April 21, 1999 AMS Subject Classification: 05A17, 05A19, 05A30, 11P81 Abstract. We continue the study of the method outlined by MacMahon in Section VIII of [10]. The long range object is to show the relevance of MacMahon's ideas in current partition-theoretic research. In this paper we present a number of theorems which MacMahon overlooked. For example, the number of partitions of n with non-negative first and second differences between parts equals the number of partitions of n into triangular numbers. Keywords: Partition Analysis, partitions, plane partitions References 1. G.E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly 86 (1979) 477–478. 2. G.E. Andrews, MacMahon’s partition analysis: I. The lecture hall partition theorem, In: Mathematical Essays in Honor Gian-Carlo Rota, B. Sagan and R. Stanley, Eds., Birkhäuser, Boston, 1998, pp. 1–22. 3. G.E. Andrews, P. Paule, and A. Riese, MacMahon’s partition analysis: III. The omega package, to appear. 4. G.E. Andrews and P. Paule, MacMahon’s partition analysis: IV. Hypergeometric multisums, to appear. 5. M. Bousquet-Mélou and K. Eriksson, Lecture hall partitions, The Ramanujan Journal 1 (1997) 101–111. 6. R. Honsberger, Mathematical Gems III, Math. Assoc. of America, Washington, 1985. 7. J.H. Jordan, R. Walsh, and R.J. Wisner, Triangles with integer sides, Notices Amer. Math. Soc. 24 (1977) A–450. 8. J.H. Jordan, R. Walsh, and R.J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (1979) 686–689. 9. C.I. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. 10. P.A. MacMahon, Combinatory Analysis, Vol. 2, Cambridge University Press, Cambridge, 1916; Chelsea, New York, 1960, reissued 11. P.A. MacMahon, Collected Papers, Vol. 1, G.E. Andrews, Ed., MIT Press, Cambridge, 1978. 12. R.P. Stanley, Linear homogeneous equations and magic labelings of graphs, Duke Math. J. 40 (1973 ) 607–632. 13. R.P. Stanley, Enumerative Combinatories, Vol. 1, Wadsworth, Monterey, 1986. |