"평면 분할 (plane partitions)"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
  
−
 
 
−
 
−
 
 
−
 
−
==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\}</math>
 
  
−
 
+
==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===
 +
$$
 +
\left\{ \begin{array}{l}  \{3\} \end{array} , \begin{array}{l}  \{2,1\} \end{array} , \begin{array}{l}  \{1,1,1\} \end{array} , \begin{array}{l}  \{2\} \\  \{1\} \end{array} , \begin{array}{l}  \{1,1\} \\  \{1\} \end{array} , \begin{array}{l}  \{1\} \\  \{1\} \\  \{1\} \end{array} \right\}
 +
$$
  
−
==3의 평면분할==
+
===4의 평면분할===
 +
$$
 +
\left\{
 +
\begin{array}{c}
 +
\{4\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{2,2\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{3,1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{2,1,1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{1,1,1,1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{2\} \\
 +
\{2\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{3\} \\
 +
\{1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{1,1\} \\
 +
\{1,1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{2,1\} \\
 +
\{1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{1,1,1\} \\
 +
\{1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{2\} \\
 +
\{1\} \\
 +
\{1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{1,1\} \\
 +
\{1\} \\
 +
\{1\} \\
 +
\end{array}
 +
,
 +
\begin{array}{c}
 +
\{1\} \\
 +
\{1\} \\
 +
\{1\} \\
 +
\{1\} \\
 +
\end{array}
 +
\right\}
 +
$$
  
−
<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\}</math>
 
−
 
−
 
 
  
 +
==생성함수==
 +
* 다음과 같이 무한곱으로 표현가능하다
 +
:<math>
 +
\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}
 +
</math>
 
 
 
 
  
57번째 줄: 129번째 줄:
  
 
==사전 형태의 자료==
 
==사전 형태의 자료==
−
* http://ko.wikipedia.org/wiki/
 
 
* http://en.wikipedia.org/wiki/Plane_partition
 
* http://en.wikipedia.org/wiki/Plane_partition
  

2014년 9월 18일 (목) 02:03 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


관련논문

  • 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.

 

원본 주소 "https://wiki.mathnt.net/index.php?title=평면_분할_(plane_partitions)&oldid=30380"