co.combinatorics - Generalization of permanent definition based on number of permutation cycles

Let $A$ be an $n$ by $n$ matrix and $x$ a free parameter. Define
$$p(A,x)=sum_{pi in S_n} x^{n(pi)}A_{1pi(1)}ldots A_{npi(n)},$$
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?

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