  We prove the conjecture by M. Yip stating that counting genus one partitions
by the number of their elements and parts yields, up to a shift of indices, the
same array of numbers as counting genus one rooted hypermonopoles. Our proof
involves representing each genus one permutation by a four-colored noncrossing
partition. This representation may be selected in a unique way for permutations
containing no trivial cycles. The conclusion follows from a general generating
function formula that holds for any class of permutations that is closed under
the removal and reinsertion of trivial cycles. Our method also provides a new
way to count rooted hypermonopoles of genus one, and puts the spotlight on a
class of genus one permutations that is invariant under an obvious extension of
the Kreweras duality map to genus one permutations.
