"평면 분할 (plane partitions)"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) (→생성함수) |
||
90번째 줄: | 90번째 줄: | ||
\sum_{\pi:\text{plane partitions}}q^{|\pi|} & = \prod_{n=1}^\infty \frac {1}{(1-q^n)^n} \\ | \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 + | & =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 | + | 282 q^9 + 500 q^{10}+\cdots |
\end{aligned} | \end{aligned} | ||
</math> | </math> |
2015년 8월 2일 (일) 23:13 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
- 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.