Editing Lah number (section)

==Rising and falling factorials==


Let <math display="inline">x^{(n)}</math> represent the [[rising factorial]] <math display="inline">x(x+1)(x+2) \cdots (x+n-1)</math> and let <math display="inline">(x)_n</math> represent the [[falling factorial]] <math display="inline">x(x-1)(x-2) \cdots (x-n+1)</math>.  The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other.  Explicitly,<math display="block">x^{(n)} = \sum_{k=0}^n L(n,k) (x)_k</math>and<math display="block">(x)_n = \sum_{k=0}^n (-1)^{n-k} L(n,k)x^{(k)}.</math>For example,<math display="block">x(x+1)(x+2) = {\color{red}6}x + {\color{red}6}x(x-1) + {\color{red}1}x(x-1)(x-2)</math>and<math display="block">x(x-1)(x-2) = {\color{red}6}x - {\color{red}6}x(x+1) + {\color{red}1}x(x+1)(x+2),</math>

where the coefficients 6, 6, and 1 are exactly the Lah numbers <math>L(3, 1)</math>, <math>L(3, 2)</math>, and <math>L(3, 3)</math>.