Séminaire Lotharingien de Combinatoire, B70e (2014), 28 pp.
Robert Cori and Gábor Hetyei
Counting Genus One Partitions and Permutations
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
Received: July 21, 2013.
Accepted: December 27, 2013.
Final Version: January 10, 2014.
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