[Some punny title about determinants and fixed points]

Prove the following identity

\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} =  \det\begin{pmatrix}  t & 1 &\cdots& 1&  1\\  1 & t &\cdots& 1& 1 \\  \vdots & \vdots &\ddots & \vdots & \vdots \\  1 & 1 &\cdots& t& 1\\  1 & 1 &\cdots& 1& t  \end{pmatrix}

the sum running through S_n and the matrix being of size n\times n.

 

 

Advertisements
[Some punny title about determinants and fixed points]