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Abstract
Recently, Andrews defined a partition function EO(n) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EO‾(n) which counts the number of partitions of n enumerated by EO(n) in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of EO‾(n). In this article, we prove infinite families of congruences for EO‾(n). We next study distribution of EO‾(n). We prove that there are infinitely many integers N in every arithmetic progression for which EO‾(2N) is even; and that there are infinitely many integers M in every arithmetic progression for which EO‾(2M) is odd so long as there is at least one. We further prove that EO‾(n) is even for almost all n. Very recently, Uncu has treated a different subset of the partitions enumerated by EO(n). We prove that Uncu's partition function is divisible by 2k for almost all k. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.
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