"평면 분할 (plane partitions)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 18개는 보이지 않습니다) | |||
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==개요== | ==개요== | ||
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| − | + | ==평면분할의 예== | |
| + | ===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> | ||
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| − | <math>\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=== |
| + | :<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> | ||
| − | + | ===4의 평면분할=== | |
| + | :<math> | ||
| + | \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> | ||
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| − | + | ==생성함수== | |
| − | + | * 다음과 같이 무한곱으로 표현가능하다 | |
| − | == | + | :<math> |
| − | + | \begin{aligned} | |
| − | <math> | + | \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> | |
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==메모== | ==메모== | ||
| 47번째 줄: | 100번째 줄: | ||
* Math Overflow http://mathoverflow.net/search?q= | * Math Overflow http://mathoverflow.net/search?q= | ||
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==관련된 항목들== | ==관련된 항목들== | ||
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==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== | ||
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMGI2OTE4NTMtYWMzZS00OWZjLTliYTgtZThiMjM2YmY2ZTg5&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxMGI2OTE4NTMtYWMzZS00OWZjLTliYTgtZThiMjM2YmY2ZTg5&sort=name&layout=list&num=50 | ||
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| − | + | ||
| − | + | ==사전 형태의 자료== | |
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| − | ==사전 | ||
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* http://en.wikipedia.org/wiki/Plane_partition | * http://en.wikipedia.org/wiki/Plane_partition | ||
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| − | + | ==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. | ||
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==관련논문== | ==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. | ||
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| − | + | [[분류:q-급수]] | |
| − | + | [[분류:분할수]] | |
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| − | * | + | ==메타데이터== |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q7201015 Q7201015] |
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'plane'}, {'LEMMA': 'partition'}] | ||
2021년 2월 17일 (수) 05:06 기준 최신판
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
\[ \begin{aligned} \sum_{\pi:\text{plane partitions}}q^{|\pi|} & = \prod_{n=1}^\infty \frac {1}{(1-q^n)^n} \\ & =1 + q + 3 q^2 + 6 q^3 + 13 q^4 + 24 q^5 + 48 q^6 + 86 q^7 + 160 q^8 + 282 q^9 + 500 q^{10}+\cdots \end{aligned} \]
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.
메타데이터
위키데이터
- ID : Q7201015
Spacy 패턴 목록
- [{'LOWER': 'plane'}, {'LEMMA': 'partition'}]