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We prove the conjecture by M. Yip stating that counting genus one partitionsby the number of their elements and parts yields, up to a shift of indices, thesame array of numbers as counting genus one rooted hypermonopoles. Our proofinvolves representing each genus one permutation by a four-colored noncrossingpartition. This representation may be selected in a unique way for permutationscontaining no trivial cycles. The conclusion follows from a general generatingfunction formula that holds for any class of permutations that is closed underthe removal and reinsertion of trivial cycles. Our method also provides a newway to count rooted hypermonopoles of genus one, and puts the spotlight on aclass of genus one permutations that is invariant under an obvious extension ofthe Kreweras duality map to genus one permutations.

Counting genus one partitions and permutations