It would lead me too far to explain how I stumbled upon the somewhat obscure identities $$\sum_{m=0}^n \binom{n}{m} (1-m)^m m^{n-m}=(-1)^n d_n, \quad \sum_{m=0}^n \binom{n}{m} (1-m)^{m-1} m^{n-m}=1,$$ where $d_n=\sum^{n}_{k=0} (-1)^k /k!$ is the $n$-th de Montmort number, when doing some algebro-geometric considerations. No doubt they are well-known to professional combinatorialists. As stated, they look rather unmotivated; surely they are special cases of a much broader class of identities. Ideally I would like to get a reference for the latter.

It would lead me too far to explain how I stumbled upon the somewhat obscure identities $$\sum_{m=0}^n \binom{n}{m} (1-m)^m m^{n-m}=(-1)^n d_n, \quad \sum_{m=0}^n \binom{n}{m} (1-m)^{m-1} m^{n-m}=1,$$ where $d_n=n!\sum^{n}_{k=0} (-1)^k /k!$ is the $n$-th de Montmort number, when doing some algebro-geometric considerations. No doubt they are well-known to professional combinatorialists. As stated, they look rather unmotivated; surely they are special cases of a much broader class of identities. Ideally I would like to get a reference for the latter.