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.



A certain type of combinatorial identity, involving de Montmort numbers
In John Riordan's book *Combinatorial Identities*, page 21, is the formula
$$\sum\_{k=0}^n \binom{n}{k}(x+k)^k (y+n-k)^{n-k}=
 \sum\_{k=0}^n \binom{n}{k} k!\, (x+y+n)^{n-k}.$$
(There is a typo in the formula given in the book—this is the correct version.) Riordan writes, "This is usually called Cauchy's formula," but I don't know of a reference to Cauchy's work. The case $x=-1, y=-n$ is the OP's first formula.


Sections 1.5 and 1.6 of Riordan's book are devoted to Abel-type formulas.


