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Bell polynomials : ウィキペディア英語版
Bell polynomials

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are a triangular array of polynomials given by
:B_(x_1,x_2,\dots,x_)
:=\sum
\left(\right)^\left(\right)^\cdots\left(\right)^= k\quad\mbox\quad j_1+2j_2+3j_3+\cdots + (n-k+1)j_=n.
==Complete Bell polynomials==

The sum
:B_n(x_1,\dots,x_n)=\sum_^n B_(x_1,x_2,\dots,x_)
is sometimes called the ''n''th complete Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials ''B''''n'',''k'' are sometimes called ''partial'' or ''incomplete'' Bell polynomials.
The complete Bell polynomials satisfy the following identity
:B_n(x_1,\dots,x_n) = \det\beginx_1 & x_2 & x_3 & x_4 & x_5 & \cdots & \cdots & x_n \\ \\
-1 & x_1 & x_2 & x_3 & x_4 & \cdots & \cdots & x_ \\ \\
0 & -1 & x_1 & x_2 & x_3 & \cdots & \cdots & x_ \\ \\
0 & 0 & -1 & x_1 & x_2 & \cdots & \cdots & x_ \\ \\
0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_ \\ \\
0 & 0 & 0 & 0 & -1 & \cdots & \cdots & x_ \\ \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \\
0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1 \end.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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