Abstract
Two disconnected remarks about partitions. First, a pedagogical remark connecting pure mathematics with statistical physics. The grand canonical ensemble of statistical mechanics is applied to the counting of partitions. This picture borrowed from physics gives a simple approximation to the exact calculation of the partition function by Hardy and Ramanujan. Second, an exact formula is guessed for the function N S (m,n) defined in a recent paper by Andrews, Garvan, and Liang. The formula was subsequently proved by Garvan. We hope that it may lead to a better understanding of the beautiful new congruence properties of partitions discovered by Andrews.
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Andrews, G.E., Garvan, F.G., Liang, J.L.: Combinatorial interpretations of congruences for the SPT-function. Ramanujan J. (this issue)
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc. 305, 47–77 (1988)
Garvan, F.G.: private communication
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. (2) 17, 75–115 (1918)
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Dyson, F. Partitions and the grand canonical ensemble. Ramanujan J 29, 423–429 (2012). https://doi.org/10.1007/s11139-012-9387-5
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DOI: https://doi.org/10.1007/s11139-012-9387-5