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Séminaire Lotharingien de Combinatoire, B70e (2014), 28 pp.

# Robert Cori and Gábor Hetyei

# Counting Genus One Partitions and Permutations

**Abstract.**
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.

Received: July 21, 2013.
Accepted: December 27, 2013.
Final Version: January 10, 2014.

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