평면 분할 (plane partitions)

개요

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

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

역사

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