 Short-cycle expectation in a uniform regular graph

For fixed $r,\ell \ge 3$ and $rn$ even, a prescribed graph cycle in the configuration model has two ordered distinct half-edges at each of its vertices. Therefore its expected count is $(n{)}_{\ell }[r(r-1){]}^{\ell }/[2\ell {\prod }_{j=0}^{\ell -1}(rn-2j-1)]$. Conditioning on the chosen cycle leaves degrees $r-2$ at its vertices and $r$ elsewhere; forbid further edges along that cycle. The bounded-degree pairing avoidance estimate changes by $o(1)$, so the simplicity probability conditional on the cycle divided by the unconditional probability tends to one. The displayed expectation therefore holds for the uniform simple graph, not just the unconditioned pairing model.