A continuous family of partition statistics equidistributed with length

AbstractThis article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product ∏i⩾11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees

Cylindrical lattice walks and the Loehr–Warrington 10n conjecture

AbstractThe following special case of a conjecture by Loehr and Warrington was proved recently by Ekhad, Vatter, and Zeilberger:There are 10n zero-sum words of length 5n in the alphabet {+3,−2} such that no zero-sum consecutive subword that starts with +3 may be followed immediately by −2.We give a simple bijective proof of the conjecture in its original and more general setting. To do this we reformulate the problem in terms of cylindrical lattice walks