Laboratoire de Physique Théorique
Toulouse - UMR 5152

***NOTE UNUSUAL TIME****

Partitions of integers appear in a large number of areas such as number theory, combinatorics, statistical physics and string theory. MacMahon defined higher dimensional (d>1) generalizations of the usual (d=1) partitions and conjectured generating functions for them and proved his conjecture for d=2. However, the MacMahon conjecture fails for d>2. Knuth, in 1970, published a paper addressing the problem of efficient enumeration of solid (d=3) partitions. We will show some improvements to this method that gives a speed-up of more than 10000x and there is scope to increase the speed-up. This, along with parallelization of the algorithm allowed us to add 18 more terms corresponding to N=51 to N=68. This improvement can be generalized to d>3 and for d=3,4,5 significantly more terms have been added. It has been conjectured by Mustonen and Rajesh that though the MacMahon conjecture for d=3 is not true, it may explain the asymptotic behaviour of these numbers. Our work appears to enhance the numerical evidence for this conjecture. In this seminar, we will be seeing the improved version of the exact algorithm for enumeration of 3d partition, the techniques involved in the proof of generating functions of 2d partitions and conjectures related to the asymptotics of higher dimensional partitions that arose from our work.

References :

S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, 2012, On the asymptotics of higher dimensional partitions, J.Phys. A45 (2012) 055001 arXiv:1105.6231