"Birkhoff–von Neumann polytope"의 두 판 사이의 차이
imported>Pythagoras0 (새 문서: ==computational resource== * https://drive.google.com/file/d/0B8XXo8Tve1cxRDRnNGlwcGZLN0E/view * http://www.math.binghamton.edu/dennis/Birkhoff/) |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 6개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| + | ==introduction== | ||
| + | A magic square is a square matrix with nonnegative integer entries | ||
| + | having all line sums equal to each other, where a line is a row or a column. | ||
| + | Let <math>H_n (r)</math> be the number of <math>n \times n</math> magic squares with line sums equal | ||
| + | to <math>r</math>. The problem to determine <math>H_n (r)</math> appeared early in the twentieth | ||
| + | century \cite{Ma}. Since then it has attracted considerable attention | ||
| + | within areas such as combinatorics, combinatorial and computational | ||
| + | commutative algebra, discrete and computational geometry, probability | ||
| + | and statistics \cite{ADG, BP, DG, Eh, JvR, Sp, St1, St2, St4, St5, SS}. | ||
| + | It was conjectured by Anand, Dumir and Gupta \cite{ADG} and proved by | ||
| + | Ehrhart \cite{Eh} and Stanley \cite{St1} (see also | ||
| + | \cite[Section I.5]{St4} and \cite[Section 4.6]{St5}) that for any fixed | ||
| + | positive integer <math>n</math>, the quantity <math>H_n (r)</math> is a polynomial in <math>r</math> of | ||
| + | degree <math>(n-1)^2</math>. More precisely, the following theorem holds. | ||
| + | |||
| + | ;begin{theorem} {\rm (Stanley~\cite{St1, St2})} | ||
| + | For any positive integer <math>n</math> we have | ||
| + | |||
| + | \begin{equation} | ||
| + | \sum_{r \ge 0} \, H_n (r) \, t^r = \frac{h_0 + h_1 t + \cdots + h_d t^d} | ||
| + | {(1 - t)^{(n-1)^2 + 1}}, | ||
| + | \label{mag0} | ||
| + | \end{equation} | ||
| + | |||
| + | where <math>d = n^2 - 3n + 2</math> and the <math>h_i</math> are nonnegative integers satisfying | ||
| + | <math>h_0 = 1</math> and <math>h_i = h_{d-i}</math> for all <math>i</math>. | ||
| + | \label{thm0} | ||
| + | |||
| + | |||
| + | It is the first conjecture stated in \cite{St4} (see Section I.1 there) that | ||
| + | the integers <math>h_i</math> appearing in (\ref{mag0}) satisfy further the inequalities | ||
| + | |||
| + | \begin{equation} | ||
| + | h_0 \le h_1 \le \cdots \le h_{\lfloor d/2 \rfloor}. | ||
| + | \label{mag1} | ||
| + | \end{equation} | ||
| + | |||
| + | |||
| + | ==expositions== | ||
| + | * http://www.math.binghamton.edu/dennis/Birkhoff/ | ||
| + | |||
==computational resource== | ==computational resource== | ||
* https://drive.google.com/file/d/0B8XXo8Tve1cxRDRnNGlwcGZLN0E/view | * https://drive.google.com/file/d/0B8XXo8Tve1cxRDRnNGlwcGZLN0E/view | ||
| − | * http://www.math.binghamton.edu/dennis/Birkhoff/ | + | * http://www.math.binghamton.edu/dennis/Birkhoff/polynomials.html |
| + | |||
| + | == articles == | ||
| + | |||
| + | * Christos A. Athanasiadis, Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, arXiv:math/0312031 [math.CO], December 01 2003, http://arxiv.org/abs/math/0312031 | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q4916482 Q4916482] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'birkhoff'}, {'LEMMA': 'polytope'}] | ||
2021년 2월 17일 (수) 02:01 기준 최신판
introduction
A magic square is a square matrix with nonnegative integer entries having all line sums equal to each other, where a line is a row or a column. Let \(H_n (r)\) be the number of \(n \times n\) magic squares with line sums equal to \(r\). The problem to determine \(H_n (r)\) appeared early in the twentieth century \cite{Ma}. Since then it has attracted considerable attention within areas such as combinatorics, combinatorial and computational commutative algebra, discrete and computational geometry, probability and statistics \cite{ADG, BP, DG, Eh, JvR, Sp, St1, St2, St4, St5, SS}. It was conjectured by Anand, Dumir and Gupta \cite{ADG} and proved by Ehrhart \cite{Eh} and Stanley \cite{St1} (see also \cite[Section I.5]{St4} and \cite[Section 4.6]{St5}) that for any fixed positive integer \(n\), the quantity \(H_n (r)\) is a polynomial in \(r\) of degree \((n-1)^2\). More precisely, the following theorem holds.
- begin{theorem} {\rm (Stanley~\cite{St1, St2})}
For any positive integer \(n\) we have
\begin{equation} \sum_{r \ge 0} \, H_n (r) \, t^r = \frac{h_0 + h_1 t + \cdots + h_d t^d} {(1 - t)^{(n-1)^2 + 1}}, \label{mag0} \end{equation}
where \(d = n^2 - 3n + 2\) and the \(h_i\) are nonnegative integers satisfying \(h_0 = 1\) and \(h_i = h_{d-i}\) for all \(i\). \label{thm0}
It is the first conjecture stated in \cite{St4} (see Section I.1 there) that
the integers \(h_i\) appearing in (\ref{mag0}) satisfy further the inequalities
\begin{equation} h_0 \le h_1 \le \cdots \le h_{\lfloor d/2 \rfloor}. \label{mag1} \end{equation}
expositions
computational resource
- https://drive.google.com/file/d/0B8XXo8Tve1cxRDRnNGlwcGZLN0E/view
- http://www.math.binghamton.edu/dennis/Birkhoff/polynomials.html
articles
- Christos A. Athanasiadis, Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, arXiv:math/0312031 [math.CO], December 01 2003, http://arxiv.org/abs/math/0312031
메타데이터
위키데이터
- ID : Q4916482
Spacy 패턴 목록
- [{'LOWER': 'birkhoff'}, {'LEMMA': 'polytope'}]