Plane partitions

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An ordinary (i.e. one-dimensional) partition is a row of nonnegative integers in nonincreasing (weakly decreasing) order whose sum is $\scriptstyle n\,$.

A plane partition of $\scriptstyle n\,$ is a two-dimensional arrangement of nonnegative integers

$n_{0,0}\,$	$n_{0,1}\,$	$n_{0,2}\,$	$n_{0,3}\,$	$\cdots$
$n_{1,0}\,$	$n_{1,1}\,$	$n_{1,2}\,$	$n_{1,3}\,$	$\cdots$
$n_{2,0}\,$	$n_{2,1}\,$	$n_{2,2}\,$	$n_{2,3}\,$	$\cdots$
$n_{3,0}\,$	$n_{3,1}\,$	$n_{3,2}\,$	$n_{3,3}\,$	$\cdots$
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$

which are nonincreasing (weakly decreasing) in both rows and columns:

$n_{i+1,j}\leq n_{i,j},\ n_{i,j+1}\leq n_{i,j}\,$

and which sum to $\scriptstyle n\,$:

$n=\sum _{i,j}n_{i,j}.\,$

Number of plane partitions

The number of planar partitions of $\scriptstyle n,\,n\,\geq \,0,\,$ gives the sequence (Cf. A000219)

{1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, ...}

MacMahon's formula

MacMahon (1960) showed that the number $\scriptstyle M(a,b,c)\,$ of plane partitions whose Young diagrams fit inside an $\scriptstyle a\times b\,$ rectangle and whose integers do not exceed $\scriptstyle c\,$ (with all $\scriptstyle n_{i,j}\,\leq \,c\,$) is given by

$M(a,b,c)=\prod _{i=1}^{a}\prod _{j=1}^{b}\prod _{k=1}^{c}{\frac {i+j+k-1}{i+j+k-2}}.$

This formula was obtained by Percy MacMahon and was later rewritten in this form by Ian Macdonald.

Recurrence

$p_{2}(n)={\frac {1}{n}}\sum _{k=1}^{n}p_{2}(n-k)\sigma _{2}(k),\,$

where $\scriptstyle \sigma _{2}(n)\,$ is the sum of squares of divisors of $\scriptstyle n\,$ (cf. divisor function.)

Generating function

The generating function of ordinary (one-dimensional) partitions is the reciprocal of Euler's function

$G_{\{p_{1}(n)\}}(x)\equiv \sum _{n=0}^{\infty }p_{1}(n)x^{n}={\frac {1}{(1-x)^{1}(1-x^{2})^{1}(1-x^{3})^{1}(1-x^{4})^{1}\cdots }}={\frac {1}{\prod _{n=1}^{\infty }(1-x^{n})}}.\,$

In 1912, Major Percy A. MacMahon proved that the generating function for plane partitions is

$G_{\{p_{2}(n)\}}(x)\equiv \sum _{n=0}^{\infty }p_{2}(n)x^{n}={\frac {1}{(1-x)^{1}(1-x^{2})^{2}(1-x^{3})^{3}(1-x^{4})^{4}\cdots }}={\frac {1}{\prod _{n=1}^{\infty }(1-x^{n})^{n}}}=1+x+3x^{2}+6x^{3}+13x^{4}+24x^{5}+\ldots \,$

Now, you may be tempted to conjecture the generating function for solid partitions...

Also

$G_{\{p_{2}(n)\}}(x)\equiv \sum _{n=0}^{\infty }p_{2}(n)x^{n}=\exp {{\Bigg [}\sum _{n=1}^{\infty }\sigma _{2}(n){\frac {x^{n}}{n}}{\Bigg ]}},\,$

where $\scriptstyle \sigma _{2}(n)\,$ is the sum of squares of divisors of $\scriptstyle n\,$ (cf. divisor function.)

Asymptotic behaviour

...

See also

• Partitions (one-dimensional partitions)
• Plane partitions (two-dimensional partitions)
• Solid partitions (three-dimensional partitions)

• Compositions (one-dimensional ordered partitions)
• Plane compositions (two-dimensional ordered partitions)
• Solid compositions (three-dimensional ordered partitions)

• Multiplicative partitions (factorizations)
• Multiplicative compositions (ordered factorizations)

• Prime signature (which may be viewed as a given partition of $\scriptstyle \Omega (n)\,$, the number of prime factors (with repetition) of $\scriptstyle n\,$.)

External links

• Donald E. Knuth, A Note on Solid Partitions, 1970.
• E. Maitland Wright, ASYMPTOTIC PARTITION FORMULAE, Q J Math (1931) os-2 (1): 177-189.
• Ljuben R. Mutafchiev, Asymptotic Enumeration of Plane Partitions of Large Integers and Hayman’s Theorem for Admissible Generating Functions, American University in Bulgaria.
• Weisstein, Eric W., Plane Partition, from MathWorld—A Wolfram Web Resource..