University of Minnesota Combinatorics Seminar
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Abstract |
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Pak and Stanley constructed a map from the set of connected components of the complement to the k-Shi arrangement to the set of k-parking functions. It follows from the work of Fishel and Vazirani that these connected components are in bijection with kn+1-stable affine permutations on n elements (i.e. permutations with no inversions of height kn+1). We generalize Pak-Stanley labeling by constructing a map from the set of m-stable affine permutations to the set of rational slope parking functions. It follows from the work of Stanley that the map is a bijection for m=kn+1. We extend this argument to cover the case m=kn-1 and conjecture that the map is a bijection for all relatively prime (m,n). We also show that m-stable permutations label the cells in a cell decomposition in an affine Springer fiber considered by Hikita, with the dimension of the corresponding cell equal to the sum of values of the corresponding parking function.
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