평면 분할 (plane partitions)

평면분할의 예

2의 평면분할 목록

\[ \left\{ \begin{array}{l} \{2\} \end{array} , \begin{array}{l} \{1,1\} \end{array} , \begin{array}{l} \{1\} \\ \{1\} \end{array} \right\} \]

3의 평면분할

\[ \left\{ \begin{array}{l} \{3\} \end{array} , \begin{array}{l} \{2,1\} \end{array} , \begin{array}{l} \{1,1,1\} \end{array} , \begin{array}{l} \{2\} \\ \{1\} \end{array} , \begin{array}{l} \{1,1\} \\ \{1\} \end{array} , \begin{array}{l} \{1\} \\ \{1\} \\ \{1\} \end{array} \right\} \]

4의 평면분할

\[ \left\{ \begin{array}{c} \{4\} \\ \end{array} , \begin{array}{c} \{2,2\} \\ \end{array} , \begin{array}{c} \{3,1\} \\ \end{array} , \begin{array}{c} \{2,1,1\} \\ \end{array} , \begin{array}{c} \{1,1,1,1\} \\ \end{array} , \begin{array}{c} \{2\} \\ \{2\} \\ \end{array} , \begin{array}{c} \{3\} \\ \{1\} \\ \end{array} , \begin{array}{c} \{1,1\} \\ \{1,1\} \\ \end{array} , \begin{array}{c} \{2,1\} \\ \{1\} \\ \end{array} , \begin{array}{c} \{1,1,1\} \\ \{1\} \\ \end{array} , \begin{array}{c} \{2\} \\ \{1\} \\ \{1\} \\ \end{array} , \begin{array}{c} \{1,1\} \\ \{1\} \\ \{1\} \\ \end{array} , \begin{array}{c} \{1\} \\ \{1\} \\ \{1\} \\ \{1\} \\ \end{array} \right\} \]

생성함수

• 다음과 같이 무한곱으로 표현가능하다

\[ \begin{aligned} \sum_{\pi:\text{plane partitions}}q^{|\pi|} & = \prod_{n=1}^\infty \frac {1}{(1-q^n)^n} \\ & =1 + q + 3 q^2 + 6 q^3 + 13 q^4 + 24 q^5 + 48 q^6 + 86 q^7 + 160 q^8 + 282 q^9 + 500 q^{10}+\cdots \end{aligned} \]

메모

• http://users.telenet.be/Wouter.Meeussen/?C=M;O=A
• Math Overflow http://mathoverflow.net/search?q=

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxMGI2OTE4NTMtYWMzZS00OWZjLTliYTgtZThiMjM2YmY2ZTg5&sort=name&layout=list&num=50

사전 형태의 자료

• http://en.wikipedia.org/wiki/Plane_partition

리뷰, 에세이, 강의노트

• Krattenthaler, C. ‘Plane Partitions in the Work of Richard Stanley and His School’. arXiv:1503.05934 [math], 19 March 2015. http://arxiv.org/abs/1503.05934.

관련논문

• Andrij Rovenchak, Statistical mechanics approach in the counting of integer partitions, http://arxiv.org/abs/1603.01049v1
• Kamioka, Shuhei. “Plane Partitions with Bounded Size of Parts and Biorthogonal Polynomials.” arXiv:1508.01674 [math], August 7, 2015. http://arxiv.org/abs/1508.01674.
• Gessel, Ira M. “A Historical Survey of P-Partitions.” arXiv:1506.03508 [math], June 10, 2015. http://arxiv.org/abs/1506.03508.
• Ciucu, Mihai. ‘Four Factorization Formulas for Plane Partitions’. arXiv:1503.07915 [cond-Mat], 26 March 2015. http://arxiv.org/abs/1503.07915.
• Destainville, Nicolas, and Suresh Govindarajan. 2014. “Estimating the Asymptotics of Solid Partitions.” arXiv:1406.5605 [cond-Mat, Physics:hep-Th], June. http://arxiv.org/abs/1406.5605.