%mDimPars.tex: The number of m-Dimensional Partitions  of Eleven and Twelve
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\bf
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{
The number of $m$-Dimensional Partitions of Eleven and Twelve
}
\rm
\bigskip
\centerline{ {\it Shalosh B. EKHAD }\footnote{$^1$}
{\eightrm  \raggedright
%\break
{\eighttt ShaloshBEkhad@gmail.com} \quad .
\hfill \break
Accompanied by Maple package {\eighttt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/mDimPars}
written by D. Zeilberger .
\hfill \break
Exclusively published in: {\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html}  . \hfill \break
Feb. 15, 2012.
}
}


Way back in the sixties, when computation was slow and difficult, in a true {\it tour de force},
Oliver Atkin (of {\it Ramanujan congruences} fame), Ian Macdonald (of {\it Symmetric Functions} and {\it Macdonald Polynomials} fame),
John McKay (of {\it moonshine} fame), together with the {\it not-as-famous-but-nevertheless-computer-whiz} Paul Bratley,
disproved an unfortunate conjecture of Major Percy Alexander MacMahon of {\it 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 $1 \leq n \leq 10$. 
Their work was beautifully, and leisurely,  described by {\it Partitions guru} George Andrews ([A], ch. 11.).
$M_m(n)$ agreed with MacMahon's conjecture (see [A]) for $n \leq 5$, 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 $n \leq 12$. 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 $n \leq 12$ 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 $n \leq 10$.


{\bf Theorem}: Let $M_m(n)$ be the number of $m$-dimensional partitions of $n$, then
$$
M_m(1)=1 \quad ,
$$
$$
M_m(2)=m+1 \quad ,
$$
$$
M_m(3)=  1/2\, \left( m+2 \right)  \left( m+1 \right)
\quad ,
$$
$$
M_m(4)= 1/6\, \left( m+1 \right)  \left( {m}^{2}+8\,m+6 \right)
\quad ,
$$
$$
M_m(5)= 1/24\, \left( m+1 \right)  \left( {m}^{3}+21\,{m}^{2}+38\,m+24 \right)
\quad ,
$$
$$
M_m(6)= {\frac {1}{120}}\, \left( m+4 \right)  \left( m+1 \right)  \left( {m}^{3}+40\,{m}^{2}+61\,m+30 \right)
\quad ,
$$
$$
M_m(7)= {\frac {1}{720}}\, \left( m+1 \right)  \left( {m}^{5}+80\,{m}^{4}+965\,{m}^{3}+1810\,{m}^{2}+1824\,m+720 \right)
\quad ,
$$
$$
M_m(8)= {\frac {1}{5040}}\, \left( m+1 \right)  \left( {m}^{6}+132\,{m}^{5}+3235\,{m}^{4}+13320\,{m}^{3}+16864\,{m}^{2}+16848\,m+5040 \right)
\quad ,
$$
$$
M_m(9)= 
{\frac {1}{40320}}\, \left( m+1 \right)  \cdot
$$
$$
\left( {m}^{7}+203\,{m}^{6}+8911\,{m}^{5}+82145\,{m}^{4}+125944\,{m}^{3}+217532\,{m}^{2}+129744\,m+40320
 \right)
\quad ,
$$
$$
M_m(10)={\frac {1}{362880}}\, \left( m+1 \right)  \cdot
$$
$$
\left( {m}^{8}+296\,{m}^{7}+21238\,{m}^{6}+387884\,{m}^{5}+1273069\,{m}^{4}+1377404\,{m}^{3}+3261852\,{m}^{2}+
935856\,m+362880 \right)  
\quad ,
$$
$$
M_m(11)= {\frac {1}{3628800}}\, \left( m+1 \right) \cdot
$$
$$
  ( {m}^{9}+414\,{m}^{8}+45366\,{m}^{7}+1464624\,{m}^{6}+10983189\,{m}^{5}+8292186\,{m}^{4}
$$
$$
+43268084\, {m}^{
3}+18819576\,{m}^{2}+15104160\,m+3628800 )
\quad ,
$$
$$
M_m(12)= {\frac {1}{39916800}}\, \left( m+1 \right) \cdot
$$
$$
  ( {m}^{10}+560\,{m}^{9}+88980\,{m}^{8}+4653120\,{m}^{7}+69709773\,{m}^{6}+152328120\,{m}^{5}
$$
$$
+98041670\,{
m}^{4}+1093797880\,{m}^{3}-395547624\,{m}^{2}+473807520\,m+39916800 )
\quad .
$$

{\bf 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 {\it 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 $1 \leq m \leq 13$, and this data may have already been computed  by them (and if not,
it could easily have been!))

{\bf References}


[A] G. E. Andrews, {\it ``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,
{\it Some Computations for $m$-dimensional partitions}, Proc. Cambridge Phil. Soc. {\bf 63} (1967), 1097-1100.


[BGP]
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar,  {\it On the asymptotics of higher-dimensional partitions}, arXiv:1105.6231 .

[Z] D. Zeilberger, {\tt mDimPars}, a Maple package available from \hfill\break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/mDimPars} .

\end



