벨 다항식(영어: Bell polynomial)은 조합론에서 에릭 템플 벨(Eric Temple Bell)의 이름을 따서 명명된 다항식이다. 또한 벨(Bell) 다항식은 집합 분할 연구에 사용된다. 이것은 스털링 수 및 벨 수와 관련이 있다. 그리고 이것들은 또한 파 디 브루노(Faà di Bruno)의 브루노 공식과 같은 많은 응용에서 언급된다. 벨다항식 생성함수 B n ( x ) = B n ( x 1 , … , x n ) = e x p ( − x ) ∑ k = 0 ∞ k n x k k ! {\displaystyle B_{n}(x)=B_{n}(x_{1},\ldots ,x_{n})=exp({-x})\sum _{k=0}^{\infty }{{k^{n}x^{k}} \over {k!}}} = x ∑ k = 1 n ( n − 1 k − 1 ) B k − 1 ( x ) {\displaystyle \;\;\;=x\sum _{k=1}^{n}{n-1 \choose k-1}B_{k-1}(x)} Remove ads초기 벨 다항식 B 0 = 1 B 1 ( x 1 ) = x 1 B 2 ( x 1 , x 2 ) = x 1 2 + x 2 B 3 ( x 1 , x 2 , x 3 ) = x 1 3 + 3 x 1 x 2 + x 3 B 4 ( x 1 , x 2 , x 3 , x 4 ) = x 1 4 + 6 x 1 2 x 2 + 4 x 1 x 3 + 3 x 2 2 + x 4 B 5 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = x 1 5 + 10 x 2 x 1 3 + 15 x 2 2 x 1 + 10 x 3 x 1 2 + 10 x 3 x 2 + 5 x 4 x 1 + x 5 B 6 ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = x 1 6 + 15 x 2 x 1 4 + 20 x 3 x 1 3 + 45 x 2 2 x 1 2 + 15 x 2 3 + 60 x 3 x 2 x 1 + 15 x 4 x 1 2 + 10 x 3 2 + 15 x 4 x 2 + 6 x 5 x 1 + x 6 B 7 ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) = x 1 7 + 21 x 1 5 x 2 + 35 x 1 4 x 3 + 105 x 1 3 x 2 2 + 35 x 1 3 x 4 + 210 x 1 2 x 2 x 3 + 105 x 1 x 2 3 + 21 x 1 2 x 5 + 105 x 1 x 2 x 4 + 70 x 1 x 3 2 + 105 x 2 2 x 3 + 7 x 1 x 6 + 21 x 2 x 5 + 35 x 3 x 4 + x 7 ⋮ {\displaystyle {\begin{aligned}B_{0}={}&1\\[8pt]B_{1}(x_{1})={}&x_{1}\\[8pt]B_{2}(x_{1},x_{2})={}&x_{1}^{2}+x_{2}\\[8pt]B_{3}(x_{1},x_{2},x_{3})={}&x_{1}^{3}+3x_{1}x_{2}+x_{3}\\[8pt]B_{4}(x_{1},x_{2},x_{3},x_{4})={}&x_{1}^{4}+6x_{1}^{2}x_{2}+4x_{1}x_{3}+3x_{2}^{2}+x_{4}\\[8pt]B_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})={}&x_{1}^{5}+10x_{2}x_{1}^{3}+15x_{2}^{2}x_{1}+10x_{3}x_{1}^{2}+10x_{3}x_{2}+5x_{4}x_{1}+x_{5}\\[8pt]B_{6}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})={}&x_{1}^{6}+15x_{2}x_{1}^{4}+20x_{3}x_{1}^{3}+45x_{2}^{2}x_{1}^{2}+15x_{2}^{3}+60x_{3}x_{2}x_{1}\\&{}+15x_{4}x_{1}^{2}+10x_{3}^{2}+15x_{4}x_{2}+6x_{5}x_{1}+x_{6}\\[8pt]B_{7}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})={}&x_{1}^{7}+21x_{1}^{5}x_{2}+35x_{1}^{4}x_{3}+105x_{1}^{3}x_{2}^{2}+35x_{1}^{3}x_{4}\\&{}+210x_{1}^{2}x_{2}x_{3}+105x_{1}x_{2}^{3}+21x_{1}^{2}x_{5}+105x_{1}x_{2}x_{4}\\&{}+70x_{1}x_{3}^{2}+105x_{2}^{2}x_{3}+7x_{1}x_{6}+21x_{2}x_{5}+35x_{3}x_{4}+x_{7}\\\vdots \end{aligned}}} Remove ads미분 표현요약관점 벨 다항식은 또한 다음과 같은 미분으로 표현가능하다.[1] B n ( x 1 , … , x n ) = 1 n − 1 ( ∑ i = 2 n ∑ j = 1 i − 1 ( i − 1 ) ( i − 2 j − 1 ) x j x i − j ∂ B n − 1 ( x 1 , … , x n − 1 ) ∂ x i − 1 + ∑ i = 2 n ∑ j = 1 i − 1 x i + 1 ( i j ) ∂ 2 B n − 1 ( x 1 , … , x n − 1 ) ∂ x j ∂ x i − j + ∑ i = 2 n x i ∂ B n − 1 ( x 1 , … , x n − 1 ) ∂ x i − 1 ) {\displaystyle {\begin{aligned}B_{n}(x_{1},\ldots ,x_{n})={\frac {1}{n-1}}\left(\sum _{i=2}^{n}\right.&\sum _{j=1}^{i-1}(i-1){\binom {i-2}{j-1}}x_{j}x_{i-j}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\\[5pt]&\left.{}+\sum _{i=2}^{n}\sum _{j=1}^{i-1}{\frac {x_{i+1}}{\binom {i}{j}}}{\frac {\partial ^{2}B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{j}\partial x_{i-j}}}\right.\\[5pt]&\left.{}+\sum _{i=2}^{n}x_{i}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\right)\end{aligned}}} Remove ads행렬 표현요약관점 완전한 종 다항식 Bn 은 부분 종 다항식 Bn,k 의 합으로 표현 할 수 있습니다. B n ( x 1 , … , x n ) = ∑ k = 1 n B n , k ( x 1 , x 2 , … , x n − k + 1 ) {\displaystyle B_{n}(x_{1},\dots ,x_{n})=\sum _{k=1}^{n}B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})} 완전한 종 다항식을 행렬식으로 표현 할 수 있습니다. B n ( x 1 , … , x n ) = det [ x 1 ( n − 1 1 ) x 2 ( n − 1 2 ) x 3 ( n − 1 3 ) x 4 ( n − 1 4 ) x 5 ⋯ ⋯ x n − 1 x 1 ( n − 2 1 ) x 2 ( n − 2 2 ) x 3 ( n − 2 3 ) x 4 ⋯ ⋯ x n − 1 0 − 1 x 1 ( n − 3 1 ) x 2 ( n − 3 2 ) x 3 ⋯ ⋯ x n − 2 0 0 − 1 x 1 ( n − 4 1 ) x 2 ⋯ ⋯ x n − 3 0 0 0 − 1 x 1 ⋯ ⋯ x n − 4 0 0 0 0 − 1 ⋯ ⋯ x n − 5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 0 0 ⋯ − 1 x 1 ] {\displaystyle B_{n}(x_{1},\dots ,x_{n})=\det {\begin{bmatrix}x_{1}&{n-1 \choose 1}x_{2}&{n-1 \choose 2}x_{3}&{n-1 \choose 3}x_{4}&{n-1 \choose 4}x_{5}&\cdots &\cdots &x_{n}\\\\-1&x_{1}&{n-2 \choose 1}x_{2}&{n-2 \choose 2}x_{3}&{n-2 \choose 3}x_{4}&\cdots &\cdots &x_{n-1}\\\\0&-1&x_{1}&{n-3 \choose 1}x_{2}&{n-3 \choose 2}x_{3}&\cdots &\cdots &x_{n-2}\\\\0&0&-1&x_{1}&{n-4 \choose 1}x_{2}&\cdots &\cdots &x_{n-3}\\\\0&0&0&-1&x_{1}&\cdots &\cdots &x_{n-4}\\\\0&0&0&0&-1&\cdots &\cdots &x_{n-5}\\\\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\vdots \\\\0&0&0&0&0&\cdots &-1&x_{1}\end{bmatrix}}} Remove ads같이 보기 이항 정리 베르누이 수 케일리-헤밀턴 정리 참고 매스월드 매스월드 OEIS OEIS EOM 각주Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads
![{\displaystyle {\begin{aligned}B_{0}={}&1\\[8pt]B_{1}(x_{1})={}&x_{1}\\[8pt]B_{2}(x_{1},x_{2})={}&x_{1}^{2}+x_{2}\\[8pt]B_{3}(x_{1},x_{2},x_{3})={}&x_{1}^{3}+3x_{1}x_{2}+x_{3}\\[8pt]B_{4}(x_{1},x_{2},x_{3},x_{4})={}&x_{1}^{4}+6x_{1}^{2}x_{2}+4x_{1}x_{3}+3x_{2}^{2}+x_{4}\\[8pt]B_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})={}&x_{1}^{5}+10x_{2}x_{1}^{3}+15x_{2}^{2}x_{1}+10x_{3}x_{1}^{2}+10x_{3}x_{2}+5x_{4}x_{1}+x_{5}\\[8pt]B_{6}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})={}&x_{1}^{6}+15x_{2}x_{1}^{4}+20x_{3}x_{1}^{3}+45x_{2}^{2}x_{1}^{2}+15x_{2}^{3}+60x_{3}x_{2}x_{1}\\&{}+15x_{4}x_{1}^{2}+10x_{3}^{2}+15x_{4}x_{2}+6x_{5}x_{1}+x_{6}\\[8pt]B_{7}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})={}&x_{1}^{7}+21x_{1}^{5}x_{2}+35x_{1}^{4}x_{3}+105x_{1}^{3}x_{2}^{2}+35x_{1}^{3}x_{4}\\&{}+210x_{1}^{2}x_{2}x_{3}+105x_{1}x_{2}^{3}+21x_{1}^{2}x_{5}+105x_{1}x_{2}x_{4}\\&{}+70x_{1}x_{3}^{2}+105x_{2}^{2}x_{3}+7x_{1}x_{6}+21x_{2}x_{5}+35x_{3}x_{4}+x_{7}\\\vdots \end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ca6bd092bc775b9cba605acce7bea5e28f9a0d39)
![{\displaystyle {\begin{aligned}B_{n}(x_{1},\ldots ,x_{n})={\frac {1}{n-1}}\left(\sum _{i=2}^{n}\right.&\sum _{j=1}^{i-1}(i-1){\binom {i-2}{j-1}}x_{j}x_{i-j}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\\[5pt]&\left.{}+\sum _{i=2}^{n}\sum _{j=1}^{i-1}{\frac {x_{i+1}}{\binom {i}{j}}}{\frac {\partial ^{2}B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{j}\partial x_{i-j}}}\right.\\[5pt]&\left.{}+\sum _{i=2}^{n}x_{i}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\right)\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/80995e423dd506948d62909ee609be096c6b51ea)
