Rational generating series for affine permutation pattern avoidance

We consider the set of affine permutations that avoid a fixed permutation pattern. Crites has given a simple characterization for when this set is infinite. We find the generating series for this set using the Coxeter length statistic and prove that it can always be represented as a rational function. We also give a characterization of the patterns for which the coefficients of the generating series are periodic. The proofs exploit a new polyhedral encoding for the affine symmetric group.

Keywords

affine symmetric group, generating function, Coxeter length, permutation pattern, abacus, lattice polyhedra

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Published 9 December 2015