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J. Propp and T. Roby isolated a phenomenon in which a statistic on a set hasthe same average value over any orbit as its global average, naming ithomomesy. They proved that the cardinality statistic on order ideals of theproduct of two chains poset under rowmotion exhibits homomesy. In this paper,we prove an analogous result in the case of the product of three chains whereone chain has two elements. In order to prove this result, we generalize fromtwo to $n$ dimensions the recombination technique that D. Einstein and Proppdeveloped to study homomesy. We see that our main homomesy result does notfully generalize to an arbitrary product of three chains, nor to largerproducts of chains; however, we have a partial generalization to an arbitraryproduct of three chains. Additional corollaries include refined homomesyresults in the product of three chains and a new result on increasing tableaux.We conclude with a generalization of recombination to any ranked poset and ahomomesy result for the Type B minuscule poset cross a two element chain.

Homomesy in products of three chains and multidimensional recombination