# Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\lambda):=\begin{bmatrix} f(\lambda_1)&f(\lambda_1+1)&\cdots&f(\lambda_1+n-1)\\\ f(\lambda_2-1)&f(\lambda_2)&\cdots&f(\lambda_2+n-2)\\\ \vdots&\ddots&\ddots&\vdots\\\ f(\lambda_n-(n-1))&\cdots&f(\lambda_n-1)&f(\lambda_n)\qquad \end{bmatrix}.$$

(Note that the parts $\lambda_k$ are the arguments on the main diagonal; the partition is padded with zeros to get length $n$.)

Then an interesting identity is implied in this question: if $g(z):=\dfrac1{\Gamma(z+1)}=\dfrac1{z!}$, then we have $$\boxed{\sum\limits_{\lambda\vdash n}\det[M_{fg}(\lambda)\cdot M_{g}(\lambda)] =\dfrac {f(1)^n}{n!}=f(1)^ng(n)},$$ meaning that all other $f(k)$ for $k\ne1$ are canceled out in the summing.
The order of the $\lambda_k$'s is crucial here. Even if all partitions are ordered "backwards", i.e. increasingly, nothing cancels out.

What are references for this striking identity?