"평면 분할 (plane partitions)"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (section '관련논문' updated) |
Pythagoras0 (토론 | 기여) |
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==평면분할의 예== | ==평면분할의 예== | ||
===2의 평면분할 목록=== | ===2의 평면분할 목록=== | ||
| − | + | :<math> | |
\left\{ \begin{array}{l} \{2\} \end{array} , \begin{array}{l} \{1,1\} \end{array} , \begin{array}{l} \{1\} \\ \{1\} \end{array} \right\} | \left\{ \begin{array}{l} \{2\} \end{array} , \begin{array}{l} \{1,1\} \end{array} , \begin{array}{l} \{1\} \\ \{1\} \end{array} \right\} | ||
| − | + | </math> | |
===3의 평면분할=== | ===3의 평면분할=== | ||
| − | + | :<math> | |
\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\} | \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\} | ||
| − | + | </math> | |
===4의 평면분할=== | ===4의 평면분할=== | ||
| − | + | :<math> | |
\left\{ | \left\{ | ||
\begin{array}{c} | \begin{array}{c} | ||
| 81번째 줄: | 81번째 줄: | ||
\end{array} | \end{array} | ||
\right\} | \right\} | ||
| − | + | </math> | |
2020년 11월 13일 (금) 10:17 판
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
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 자료
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.