The number of m-Dimensional Partitions of Eleven and Twelve

Shalosh B. EKHAD ^{1}

Way back in the sixties, when computation was slow and difficult, in a true tour de force, Oliver Atkin (of Ramanujan congruences fame), Ian Macdonald (of Symmetric Functions and Macdonald Polynomials fame), John McKay (of moonshine fame), together with the not-as-famous-but-nevertheless-computer-whiz Paul Bratley, disproved an unfortunate conjecture of Major Percy Alexander MacMahon of Combinatory Analysis fame. To that end they derived explicit expressions, as polynomials in m, for M_{m}(n), the number of m-dimensional partitions of n for 1 ≤ n ≤ 10. Their work was beautifully, and leisurely, described by Partitions guru George Andrews ([A], ch. 11.). M_{m}(n) agreed with MacMahon’s conjecture (see [A]) for n ≤ 5, so M_{m}(6) sufficed to prove MacMahon wrong, and that was as far as Andrews went. But now that computers are so much faster, I asked my master, Doron Zeilberger, to spend an afternoon writing Maple code [Z] in order to extend these formulas to n = 11 and n = 12. He followed the strategy of [ABMM], but automating everything. Zeilberger didn’t try too hard to “optimize”, and using his suboptimal code, I easily got expressions for M_{m}(n) for all n ≤ 12. I am sure that if he tried a bit harder, he could have written a program that would get also n = 13, 14 and perhaps even n = 15, but who cares? I am willing to bet that no one in our lifetime would be able to compute M_{m}(300). In fact, the Riemann Hypothesis would be proven way before the exact formula for M_{m}(300) (a certain mundane polynomial of m of degree 299) would be ever known.

But since computing M_{m}(n) for n ≤ 12 was so easy, I am hereby filling a much needed gap and am stating the following extension of Theorem 11.8 of the classic [A]. For the sake of completeness I am also reproducing the previously known expressions for n ≤ 10.

Theorem: Let M_{m}(n) be the number of m-dimensional partitions of n, then

M_{m}(1) = 1 ,
M_{m}(2) = m + 1 ,
M_{m}(3) = 1/2 (m + 2) (m + 1) ,
M_{m}(4) = 1/6 (m + 1) (m^{2} + 8 m + 6) ,
M_{m}(5) = 1/24 (m + 1) (m^{3} + 21 m^{2} + 38 m + 24) ,

M_{m}(6) = \frac{1}{120} (m + 4) (m + 1) (m^{3} + 40 m^{2} + 61 m + 30) ,

1 ShaloshBEkhad@gmail.com .

Accompanied by Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/mDimPars written by D. Zeilberger .

Exclusively published in: http://www.math.rutgers.edu/~zeilberg/pj.html .

Feb. 15, 2012.

M_{m}(7) = \frac{1}{720} (m + 1) (m^{5} + 80 m^{4} + 965 m^{3} + 1810 m^{2} + 1824 m + 720) ,

M_{m}(8) = \frac{1}{5040} (m + 1) (m^{6} + 132 m^{5} + 3235 m^{4} + 13320 m^{3} + 16864 m^{2} + 16848 m + 5040) ,

M_{m}(9) = \frac{1}{40320} (m + 1) ·

(m^{7} + 203 m^{6} + 8911 m^{5} + 82145 m^{4} + 125944 m^{3} + 217532 m^{2} + 129744 m + 40320) ,

M_{m}(10) = \frac{1}{362880} (m + 1) ·

(m^{8} + 296 m^{7} + 21238 m^{6} + 387884 m^{5} + 1273069 m^{4} + 1377404 m^{3} + 3261852 m^{2} + 935856 m + 362880) ,

M_{m}(11) = \frac{1}{3628800} (m + 1) ·

(m^{9} + 414 m^{8} + 45366 m^{7} + 1464624 m^{6} + 10983189 m^{5} + 8292186 m^{4}

+43268084 m^{3} + 18819576 m^{2} + 15104160 m + 3628800) ,

M_{m}(12) = \frac{1}{39916800} (m + 1) ·

(m^{10} + 560 m^{9} + 88980 m^{8} + 4653120 m^{7} + 69709773 m^{6} + 152328120 m^{5}
+98041670 m^{4} + 1093797880 m^{3} − 395547624 m^{2} + 473807520 m + 39916800) .

Proof: Routine! (See the source code of [Z] for details).

Let me conclude by commenting that recently a renewed interest in the counting of higher-dimensional partitions (in physics!) has lead to the fascinating article [BGP]. It is possible that my “new” results are implicit there (recall that in order to decipher a polynomial of degree d it suffices to know d+1 data points, so in order to know, e.g., M_{m}(13), we only need to know M_{m}(13) for 1 ≤ m ≤ 13, and this data may have already been computed by them (and if not, it could easily have been!))

References

[A] G. E. Andrews, “The Theory of Partitions”, Addison-Wesley, 1976. Reprinted by Cambridge University Press, 1984. First paperback edition, 1998.

[ABMM] A.O.L. Atkin, P. Bratley, I.G. Macdonald, and J.K.S. McKay, Some Computations for m-dimensional partitions, Proc. Cambridge Phil. Soc. 63 (1967), 1097-1100.

[BGP] S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higherdimensional partitions, arXiv:1105.6231 .

[Z] D. Zeilberger, mDimPars, a Maple package available from

http://www.math.rutgers.edu/~zeilberg/tokhniot/mDimPars .