We study combinatorial structures arising from finite-time transition probabilities of the Totally Asymmetric Simple Exclusion Process with open boundary conditions. While much of the existing combinatorial theory regarding the TASEP concerns the steady-state distribution, we focus instead on the transient dynamics. We first show that the enumeration of transition sequences between two configurations of the open TASEP is equivalent to the enumeration of standard Young tableaux of a family of non-classical shapes which have been of recent interest in the combinatorial literature. This extends to the open-boundary setting the correspondence between the TASEP with periodic boundaries and cylindric tableaux. We then introduce a family of tableau-like objects associated with Young diagrams in which repetitions of cells are allowed, subject to the partial order induced by the diagram. For each diagram, we collect the numbers of these objects into an exponential generating function. We prove that the entries of the homogeneous open TASEP transition matrix can be expressed as signed sums of such generating functions over suitable families of diagrams. This gives a combinatorial and order-theoretic interpretation of finite-time transition probabilities for the open TASEP, analogous to the combinatorial mappings known for steady-state probabilities.