파피안(Pfaffian)

2×2 교대행렬

$\left( \begin{array}{cc} 0 & t_{1,2} \ -t_{1,2} & 0 \end{array} \right)$ 의 행렬식 $t_{1,2}^2$
4×4 교대행렬

$\left( \begin{array}{cccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} \ -t_{1,2} & 0 & t_{2,3} & t_{2,4} \ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} \ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 \end{array} \right)$ , 행렬식 $\left(t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\right){}^2$
6×6 교대행렬

$\left( \begin{array}{cccccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} & t_{1,5} & t_{1,6} \ -t_{1,2} & 0 & t_{2,3} & t_{2,4} & t_{2,5} & t_{2,6} \ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} & t_{3,5} & t_{3,6} \ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 & t_{4,5} & t_{4,6} \ -t_{1,5} & -t_{2,5} & -t_{3,5} & -t_{4,5} & 0 & t_{5,6} \ -t_{1,6} & -t_{2,6} & -t_{3,6} & -t_{4,6} & -t_{5,6} & 0 \end{array} \right)$ ,

행렬식 $\left(t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\right){}^2$

$A=(t_{i,j})$ 로 주어진 교대행렬에 대하여 파피안을 다음과 같이 정의함

$\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}t_{\sigma(2i-1),\sigma(2i)}$
n=1인 경우

$t_{1,2}$
n=2인 경우

$t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}$
n=3 인 경우

$t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}$

number of perfect matchings on a planar rectangular lattice
every non-zero term in the Pfaffian of the adjacency matrix of a graph G corresponds to a perfect matching.
통계물리에서 중요한 역할
도미노 타일링
다이머 모델
매스매티카 코드 http://en.wikipedia.org/wiki/Talk%3APfaffian#Mathematica_code
http://www.science.uva.nl/onderwijs/thesis/centraal/files/f887198315.pdf

단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
발음사전 http://www.forvo.com/search/
대한수학회 수학 학술 용어집
- http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
남·북한수학용어비교
대한수학회 수학용어한글화 게시판