Probabilistic modeling of the self-assembly of the1-dimensional DNA structures

In a recent paper, using one of the algorithmic assembly formalisms of DNA nanotechnology, we proved that one tile can self-assemble length n structures and n× n squares, which are basic shapes in the study of DNA origami. This new result within a classic Tile Assembly Model (TAM) would not have been possible without the following programming topics: how can we simulate one-dimensional staged self-assembly using the signal-passing TAM, and how can we program staged self-assembly using the available software? We provide probabilistic approaches for investigating the assembly of tile-based onedimensional structures. We obtain a probabilistic proof of Han’s hook length formula in Enumerative Combinatorics. We identify algebraic and combinatorial structures underlying these algorithmic and information theory results.