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 $k$.]