# plane partitions

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##### 1: 26.12 Plane Partitions
###### §26.12 PlanePartitions
The number of plane partitions of $n$ is denoted by $\operatorname{pp}\left(n\right)$, with $\operatorname{pp}\left(0\right)=1$. …
##### 2: 26.19 Mathematical Applications
###### §26.19 Mathematical Applications
Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …
##### 3: 26.20 Physical Applications
The latter reference also describes chemical applications of other combinatorial techniques. Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …
##### 4: 26.1 Special Notation
 $x$ real variable. … plane partition. …
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … number of plane partitions of $n$. …
##### 5: 26.21 Tables
Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\not\equiv\pm 2\pmod{5}$, partitions into parts $\not\equiv\pm 1\pmod{5}$, and unrestricted plane partitions up to 100. …
##### 6: Bibliography
• G. E. Andrews (1979) Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.
• ##### 7: Errata
• Equation (26.12.26)
26.12.26 $\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\*\exp\left(3\left(\zeta\left(3\right)% \right)^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right)$

Originally this equation was given incorrectly as

$\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right).$

Reported 2011-09-05 by Suresh Govindarajan.

• ##### 8: Bibliography M
• L. Mutafchiev and E. Kamenov (2006) Asymptotic formula for the number of plane partitions of positive integers. C. R. Acad. Bulgare Sci. 59 (4), pp. 361–366.
• ##### 9: 35.5 Bessel Functions of Matrix Argument
35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
##### 10: Bibliography L
• J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.
• O. Lehto and K. I. Virtanen (1973) Quasiconformal Mappings in the Plane. 2nd edition, Springer-Verlag, New York.
• J. Lepowsky and S. Milne (1978) Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.
• H. Levine and J. Schwinger (1948) On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.