Congruences for Restricted-Cycle and Pell-Weighted Permutations

Let $C_m(n)$ count permutations of $n$ labels whose cycle lengths arecongruent to $1$ modulo $m$, and let $Q(n)$ be the integer sequence withexponential generating function $\exp(\sum_{d\ge1}P_d x^d/d)$,where $P_d$ is the $d$-th Pell number.Classical $p$-adic integrality gives the factorial-valuation lower boundsfor $C_m(n)$ at primes $p\equiv1\pmod m$ and for $Q(n)$ at primes$p\equiv\pm1\pmod8$. These specialize to divisibility statements recorded for OEIS A193437,A193438 and A244432; the split-prime valuation bound for A244432was also recorded by Peter Bala in the OEIS entry in August 2026.We also give elementary proofs of the complementary prime-wisenon-eventual-divisibility assertions, and determine the exact$3$-adic valuation of every $Q(n)$:$v_3(Q(n))=1$ if $n\equiv2\pmod3$, and zero otherwise.The last assertion follows from an explicit anti-period modulo $9$.A related Pell-companion-weighted ordinary generating function(OEIS A204061) has a complete base-5 digit characterization modulo $5$.We further establish integrality and sharp modular properties of parameterizedPell even-power series and Fibonacci even-power series,including the OEIS conjecture families A208034 and A207969.For a further cycle-avoidance triangle (A378495), we prove a row-gcdformula using prime-cycle conjugation and a normalized shift congruence.We also establish an adjacent-column difference and describe the relevant index alignment for a related OEIS formula.