IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Counting permutations and partitions of genus one

  • Feb. 13, 2015
  • 2:30 p.m.
  • LeConte 312

Abstract

The study of counting rooted maps was initiated by Tutte, who was motivated by the four color conjecture. 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 the permutations involved. Looking at the special case of rooted hypermonopoles leads to the definition of the genus of a permutation.

In this talk we will see the proof of a conjecture by 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.

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