Let $A$ be an $n$ by $n$ matrix and $x$ a free parameter. Define $$p(A,x)=\sum_{\pi \in S_n} x^{n(\pi)}A_{1\pi(1)}\ldots A_{n\pi(n)},$$ where $\pi$ ranges over the permutation group $S_n$ and $n(\pi)$ is the number of cycles in the cycle decomposition of $\pi$. Clearly, $p(A,1)=perm(A)$, the permanent. In general, $p(A,x)$ has properties in common with the permanent such as $p(PAQ,x)=p(A,x)$ for permutation matrices $P,Q$.

Is this a well-known structure in combinatorics and where might I find more information?