Counting genus one partitions and permutations

2013/06/19 by Robert Cori, Cori, Robert, Gábor Hetyei +1 · 2 citations
Computer Science · Mathematics · #05C15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Primary 05C30 #Secondary 05C10

paper · pdf · doi:10.48550/arxiv.1306.4628

openalex publication_date 2013/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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