Combinatorial interpretation of the Schlesinger-Zudilin stuffle product

Abstract

We derive an explicit formula for the quasi--shuffle product satisfied by Schlesinger--Zudilin Multiple~$q$-Zeta Values, expressed in terms of partition data. To achieve this, we interpret Schlesinger--Zudilin Multiple~$q$-Zeta Values as generating series of distinguished marked partitions, which are partitions whose Young diagrams have certain rows and columns marked. Together with the description of duality using marked partitions in~\cite{Br2}, and Bachmanns conjecture~(\cite{BaTalk}) that all linear relations among Multiple~$q$-Zeta Values are implied by duality and the stuffle product, this paper completes the description of the conjectural structure of Multiple~$q$-Zeta Values using marked partitions.