Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the polynomial generating
function
$$
\sum\_{m=0}^N C(m)x^m = x(x+1) \dots (x+N-1),
$$
the rising factorial.

I would like a reference (or a proof) for the following.
Let $C(m,j)$ be the number of permutations (of the $N$-element set) which
have exactly $m$ cycles, $j$ of which are $1$-cycles, that is, the number
of permutations with $m$ cycles and $j$ fixed points. Then the following
should be true:
$$
\sum\_{m=0}^N \sum\_{j=0}^m C(m,j)(j-1)x^m = (N-1)\cdot (x-1)x(x+1) \dots
(x+N-2),
$$
that is, $N-1$ times the rising factorial of $x-1$.

I verified this for $N= 3,4,5,6$ by hand, and figured that if it works up
to $6$, it probably is true for all $N$ (since $S\_4$ and $S\_6$ are the
screwiest symmetric groups). But this surely must be known.

*Motivation.* Let $V$ be a $k$-dimensional vector space (over the
complexes or the rationals, it doesn't matter), form the $N$-fold tensor
product of $V$ with itself, $W = \otimes^N V$, and consider the obvious
permutation action of $S\_N$ on $W$. This is a very heavily studied
object, going back (at least) to Schur. Let $\chi$ denote the character of
this representation of $S\_N$ (this depends on $k$, of course).

It is not difficult to see that for $g \in S\_N$, we have $\chi(g) =
k^{c(g)}$ where $c(g)$ is the number of cycles in $g$. Let $\chi\_0$ be the
trivial character, and let $\chi\_s$ be the character of the standard
$N-1$-dimensional irreducible representation of $S\_N$ ($\chi\_s (g) $ is the number
of fixed points less $1$, of $g$). Then the first formula entails that
$(\chi,\chi\_0) = {{k+N-1}\choose N}$, and the second (if true) says that
$(\chi,\chi\_s) = (N-1) {{k+N-2}\choose N}$.

If $\chi\_{-}$ and $\chi\_{s-}$ are respectively the irreducible characters
obtained by multiplying $\chi\_0$ and $\chi\_s$ by the sign character (the
other linear character, denoted $\chi\_{-}$), then we also deduce from the
equations above (evaluating at $-x$ and normalizing) that $(\chi,\chi\_{-})
= {k \choose N}$ and $(\chi,\chi\_{s-}) = (N-1){{k+1} \choose N}$.

I was interested in the class function (on $S\_N$), $\psi\_k: g \mapsto
k^{c(g)}$ for $k$ positive real numbers (rather than just integers; set $x=k$); the
question was for which values of $k$ is this a *nonnegative real*
combination of irreducible characters. If the second displayed equation is
true, and $k < N-1$, then nonnegativity of $\psi\_k$ implies that $k$ is an integer.

Presumably, it is true that for all real $k > N-1$, $\psi\_k$ has only
nonnegative (and probably positive) coefficients (with respect to the
irreducible characters). I verified this for $N = 3,4$ by hand, and then
was exhausted. [A very general and easy result, using the Perron theorem
for example, is that this is true for all sufficiently large $k$.]