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Register a new userA tiling proof of Eulers pentagnal number theorem and generalizations
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In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 times infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Eulers Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Eulers Pentagonal Number Theorem along with an infinite family of generalizations.
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We obtain a unification of two refinements of Eulers partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodts insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
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