Congruences for Restricted-Cycle and Pell-Weighted Permutations

Let Cₘ(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(∑d≥1Pd xᵈ/d),where Pd is the d-th Pell number.Classical p-adic integrality gives the factorial-valuation lower boundsfor Cₘ(n) at primes p≡1 m and for $Q(n)$ at primesp≡±18. 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 $Q(n)$:v₃(Q(n))=1 if n≡23, 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.