Solid partition explained

In mathematics, solid partitions are natural generalizations of integer partitions and plane partitions defined by Percy Alexander MacMahon.[1] A solid partition of

n

is a three-dimensional array of non-negative integers

ni,j,k

(with indices

i,j,k\geq1

) such that

\sumi,j,kni,j,k=n

and

ni+1,j,k\leqni,j,k, ni,j+1,k\leqni,j,kand ni,j,k+1\leqni,j,k

for all

i,jandk.

Let

p3(n)

denote the number of solid partitions of

n

. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.[2]

Ferrers diagrams for solid partitions

Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of

n

is a collection of

n

points or nodes,

λ=(y1,y2,\ldots,yn)

, with

yi\in

4
Z
\geq0
satisfying the condition:[3]

Condition FD: If the node

a=(a1,a2,a3,a4)\inλ

, then so do all the nodes

y=(y1,y2,y3,y4)

with

0\leqyi\leqai

for all

i=1,2,3,4

.

For instance, the Ferrers diagram

\left(\begin{smallmatrix}0\ 0\0\ 0\end{smallmatrix} \begin{smallmatrix}0\ 0\1\ 0\end{smallmatrix} \begin{smallmatrix}0\ 1\ 0\ 0\end{smallmatrix} \begin{smallmatrix}1\ 0\ 0\ 0\end{smallmatrix} \begin{smallmatrix}1\ 1\0\ 0\end{smallmatrix} \right),

where each column is a node, represents a solid partition of

5

. There is a natural action of the permutation group

S4

on a Ferrers diagram – this corresponds to permuting the four coordinates of all nodes. This generalises the operation denoted by conjugation on usual partitions.

Equivalence of the two representations

Given a Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.

Let

ni,j,k

be the number of nodes in the Ferrers diagram with coordinates of the form

(i-1,j-1,k-1,*)

where

*

denotes an arbitrary value. The collection

ni,j,k

form a solid partition. One can verify that condition FD implies that the conditions for a solid partition are satisfied.

Given a set of

ni,j,k

that form a solid partition, one obtains the corresponding Ferrers diagram as follows.

Start with the Ferrers diagram with no nodes. For every non-zero

ni,j,k

, add

ni,j,k

nodes

(i-1,j-1,k-1,y4)

for

0\leqy4<ni,j,k

to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.

For example, the Ferrers diagram with

5

nodes given above corresponds to the solid partition with

n1,1,1=n2,1,1=n1,2,1=n1,1,2=n2,2,1=1

with all other

ni,j,k

vanishing.

Generating function

Let

p3(0)\equiv1

. Define the generating function of solid partitions,

P3(q)

, by

P3(q)

infty
:=\sum
n=0

p3(n)qn=1+q+4q2+10q3+26q4+59q5+140q6+.

The generating functions of integer partitions and plane partitions have simple product formulae, due to Euler and MacMahon, respectively. However, a guess of MacMahon fails to correctly reproduce the solid partitions of 6. It appears that there is no simple formula for the generating function of solid partitions; in particular, there cannot be any formula analogous to the product formulas of Euler and MacMahon.[4]

Exact enumeration using computers

Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically. There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations. The work of Atkin. et al. used an algorithm due to Bratley and McKay.[5] In 1970, Knuth proposed a different algorithm to enumerate topological sequences that he used to evaluate numbers of solid partitions of all integers

n\leq28

.[6] Mustonen and Rajesh extended the enumeration for all integers

n\leq50

.[7] In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers

n\leq72

.[8] One finds

p3(72)=3464274974065172792 ,

which is a 19 digit number illustrating the difficulty in carrying out such exact enumerations.

Asymptotic behavior

It is conjectured that there exists a constant

c

such that[9] [10]

\lim_ \frac = c.

External links

Notes and References

  1. P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
  2. G. E. Andrews, The theory of partitions, Cambridge University Press, 1998.
  3. A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.
  4. Book: Stanley, Richard P.. Richard P. Stanley. Enumerative Combinatorics, volume 2. Cambridge University Press. 1999. 402.
  5. P. Bratley and J. K. S. McKay, "Algorithm 313: Multi-dimensional partition generator", Comm. ACM, 10 (Issue 10, 1967), p. 666.
  6. D. E. Knuth, "A note on solid partitions", Math. Comp., 24 (1970), 955–961.
  7. Ville Mustonen and R. Rajesh, "Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer", J. Phys. A: Math. Gen. 36 (2003), no. 24, 6651.cond-mat/0303607
  8. Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, "On the asymptotics of higher-dimensional partitions", J.Phys. A: Math. Gen. 45 (2012) 055001 arXiv:1105.6231.
  9. Destainville, N., & Govindarajan, S. (2015). Estimating the asymptotics of solid partitions. Journal of Statistical Physics, 158, 950-967
  10. D P Bhatia, M A Prasad and D Arora, "Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals", J. Phys. A: Math. Gen. 30 (1997) 2281