Cycle-Decorated Ribbon Complexes: Cut Coproducts and Alternating-Fence Positivity

Abstract

Let $\alpha\models n$. We define a two-variable specialization $Z_\alpha(t,q)$ of the ribbon basis of noncommutative symmetric functions from cycle enumerators of an ordinary permutation and a rooted permutation, with $q$ recording reflection length. We realize $n!Z_\alpha(t,q)$ as the shifted bigraded Euler characteristic of an $\mathfrak S_n$-equivariant ordered-set-partition complex. When $\alpha$ has at most one odd part, each total decoration determines a set of simultaneous factorization cuts, and its fiber is the classical ribbon complex indexed by that cut set. This gives explicit nonnegative ribbon expansions for every bigraded homology representation. Organizing the complexes on labelled finite sets yields a counital differential graded comonoid in the Cauchy monoidal category of species, whose cut coproduct is compatible with the fiber decomposition. At a factorization cut, the induced map on top homology is injective; its cokernel has the near-concatenation ribbon character, and the kernel of the aggregate reduced cut coproduct on homology is $H_1$. For the zigzag compositions $\delta_n$, $Z_{\delta_n}(t,-1)$ is the order polynomial of the alternating fence. We prove \[ n!Z_{\delta_n}(t,-u)\in\mathbb N[t,u] \] using coefficientwise nonnegative recurrences derived from a Riccati equation. The same cut data define a nonnegative factorization defect $d=\lceil n/2\rceil-k-j$, which controls the homological support and makes the Euler sign constant on each defect layer. We also determine the full defect-zero edge by explicit Frobenius-character formulas.