Restricted Möbius Sums on Noncrossing Partitions: Marked Matchings, Gap Factorization, and Plane Trees

Let s be a standard semicircular variable and let x be a free Poisson variable of rate one, freely independent of s. For yθ = x+ √θs, with θ≥0, we evaluate the upper-interval transform σ≥πµ(π,σ)κσ(yθ) when π has one nonsingleton block. The answer depends on the circular gaps of that block. We give explicit finite formulas in factorials and Catalan numbers, together with a proof using marked noncrossing matchings and ordered trees with binary and ternary vertices. A more general marked-matching identity applies to every base partition. It is proved by a sign-reversing involution on decorated strict chains in the original noncrossing-partition lattice. The surviving chains have both signs in general, so neither the block size nor the interval height determines a universal strict sign. Explicit examples and vanishing families illustrate the evaluation. All evaluations use finite combinatorics; no analytic transforms or formal power-series inversion are used.