Counting genus one partitions and permutations

Counting genus one partitions and permutations

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Gabor Hetyei, University of North Carolina at Charlotte
Fine Hall 214

The study of counting rooted maps was initiated by Tutte, who was motivated by the four color conjecture, and the study of hypermaps grew out of this initiative. Hypermaps are pairs of permutations that can be topologically represented by labeled maps, the genus of the underlying surface may be expressed purely algebraically in terms of these permutations. Looking at the special case of rooted hypermonopoles leads to the definition of the genus of a permutation. In this talk we prove a conjecture of Martha 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 another 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. This is joint work with Robert Cori.