In mathematics, '''solid partitions''' are natural generalizations of integer partitions and plane partitions defined by Percy Alexander MacMahon.<ref>{{cite book | last1=MacMahon | first1=P. A. | author-link1=Percy Alexander MacMahon | title=Combinatory Analysis | publisher=Cambridge University Press | place=London and New York | volume=2 | date=1916 | page=332}}</ref> A solid partition of <math> n </math> is a three-dimensional array of non-negative integers <math> n_{i,j,k}</math> (with indices <math> i, j, k\geq 1</math>) such that :<math> \sum_{i,j,k} n_{i,j,k}=n</math> and :<math> n_{i+1,j,k} \leq n_{i,j,k},\quad n_{i,j+1,k} \leq n_{i,j,k}\quad\text{and}\quad n_{i,j,k+1} \leq n_{i,j,k}</math> for all <math>i, j \text{ and } k.</math> Let <math>p_3(n)</math> denote the number of solid partitions of <math>n</math>. 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.<ref>{{cite book | last1=Andrews | first1=George E. | author-link1=George Andrews (mathematician) | title=The theory of partitions | publisher=Cambridge University Press | date=1984 | doi=10.1017/CBO9780511608650}}</ref>
== 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 <math>n</math> is a collection of <math>n</math> points or ''nodes'', <math>\lambda=(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_n)</math>, with <math>\mathbf{y}_i\in \mathbb{Z}_{\geq0}^4</math> satisfying the condition:<ref name="Atkin1967">{{cite journal | last1=Atkin | first1=A. O. L. | author-link1=A. O. L. Atkin | last2=Bratley | first2=P. | last3=McDonald | first3=I. G. | author-link3=Ian G. Macdonald | last4=McKay | first4=J. K. S. | author-link4=John McKay (mathematician) | title=Some computations for <math>m</math>-dimensional partitions | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=63 | issue=4 | date=1967 | pages=1097–1100 | doi=10.1017/S0305004100042171}}</ref>
:'''Condition FD:''' If the node <math>\mathbf{a}=(a_1,a_2,a_3, a_4)\in \lambda</math>, then so do all the nodes <math>\mathbf{y}=(y_1,y_2,y_3,y_4)</math> with <math>0\leq y_i\leq a_i</math> for all <math>i=1,2,3,4</math>.
For instance, the Ferrers diagram :<math> \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) \ , </math> where each column is a node, represents a solid partition of <math>5</math>. There is a natural action of the permutation group <math>S_4</math> 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 <math>n_{i,j,k}</math> be the number of nodes in the Ferrers diagram with coordinates of the form <math>(i-1,j-1,k-1,*)</math> where <math>*</math> denotes an arbitrary value. The collection <math>n_{i,j,k}</math> form a solid partition. One can verify that condition FD implies that the conditions for a solid partition are satisfied.
Given a set of <math>n_{i,j,k}</math> 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 <math>n_{i,j,k}</math>, add <math>n_{i,j,k}</math> nodes <math>(i-1,j-1,k-1,y_4)</math> for <math>0\leq y_4< n_{i,j,k}</math> to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.
For example, the Ferrers diagram with <math>5</math> nodes given above corresponds to the solid partition with :<math>n_{1,1,1}=n_{2,1,1}=n_{1,2,1}=n_{1,1,2}=n_{2,2,1}=1</math> with all other <math>n_{i,j,k}</math> vanishing.
== Generating function ==
Let <math>p_3(0)\equiv 1</math>. Define the generating function of solid partitions, <math>P_3(q)</math>, by :<math> P_3(q) :=\sum_{n=0}^\infty p_3(n) q^n = 1 + q + 4q^2 + 10q^3 + 26q^4 + 59q^5 + 140q^6 + \cdots </math> {{OEIS|id=A000293}}. 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.<ref name="Atkin1967"/> 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.<ref>{{cite book|last = Stanley|first = Richard P.|author-link=Richard P. Stanley|title = Enumerative Combinatorics, volume 2|publisher = Cambridge University Press| year = 1999| page = 402 | doi = 10.1017/CBO9780511609589}}</ref>
== 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.<ref>{{cite journal | last1=Bratley | first1=P. | last2=McKay | first2=J. K. S. | author-link2=John McKay (mathematician) | title=Algorithm 313: Multi-dimensional partition generator | journal=Communications of the ACM | volume=10 | issue=10 | date=1967 | page=666 | doi=10.1145/363717.363783 | doi-access=free}}</ref> In 1970, Knuth proposed a different algorithm to enumerate topological sequences that he used to evaluate numbers of solid partitions of all integers <math>n\leq 28</math>.<ref>{{cite journal | last1=Knuth | first1=Donald E. | author-link1=Donald Knuth | title=A note on solid partitions | journal=Mathematics of Computation | volume=24 | issue=112 | date=1970 | pages=955–961 | doi=10.1090/S0025-5718-1970-0277401-7 | doi-access=free}}</ref> Mustonen and Rajesh extended the enumeration for all integers <math>n\leq 50</math>.<ref name="Mustonen">{{cite journal | last1=Mustonen | first1=Ville | last2=Rajesh | first2=R. | title=Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer | journal=Journal of Physics A: Mathematical and General | volume=36 | date=2003 | issue=24 | page=6651 | arxiv=cond-mat/0303607 | doi=10.1088/0305-4470/36/24/304}}</ref> In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers <math>n\leq 72</math>.<ref>{{cite journal | last1=Balakrishnan | first1=Srivatsan | last2=Govindarajan | first2=Suresh | last3=Prabhakar | first3=Naveen S. | title=On the asymptotics of higher-dimensional partitions | journal=Journal of Physics A: Mathematical and General | volume=45 | date=2012 | article-number=055001 | arxiv=1105.6231 | doi=10.1088/1751-8113/45/5/055001}}</ref> One finds :<math> p_3(72)=3464 27497 40651 72792\ ,</math> 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 <math>c</math> such that<ref>{{cite journal | last1=Destainville | first1=Nicolas | last2=Govindarajan | first2=Suresh | date=2015 | title=Estimating the asymptotics of solid partitions | journal=Journal of Statistical Physics | volume=158 | pages=950-967 | doi=10.1007/s10955-014-1147-z| arxiv=1406.5605 }}</ref><ref name="Mustonen"/><ref>{{cite journal | last1=Bhatia | first1=D. P. | last2=Prasad | first2=M. A. | last3=Arora | first3=D. | title=Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals | journal=Journal of Physics A: Mathematical and General | volume=30 | issue=7 | date=1997 | page=2281 | doi=10.1088/0305-4470/30/7/010}}</ref>
<math display="block"> \lim_{n\rightarrow\infty} \frac{\log p_3(n)}{ n^{3/4}} = c. </math>
== References ==
<!--- See Wikipedia:Footnotes on how to create references using<ref></ref> tags which will then appear here automatically --> {{Reflist}}
== External links == * {{OEIS el|1=A000293|2=Solid (i.e., three-dimensional) partitions|formalname=a(n) = number of solid (i.e., three-dimensional) partitions of n}} * [http://boltzmann.wikidot.com/solid-partitions The Solid Partitions Project of IIT Madras] * [https://mathworld.wolfram.com/SolidPartition.html The Mathworld entry for Solid Partitions]
Category:Enumerative combinatorics Category:Integer partitions