The Mobius Function of the Permutation Pattern Poset
Abstract
A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when \sigma occurs precisely once in \tau, and \sigma and \tau satisfy certain further conditions, in which case the Mobius function is shown to be either 1, 0 or 1. We conjecture that for intervals [\sigma,\tau] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of \sigma in \tau. We also conjecture that the Mobius function of the interval [1,\tau] is 1, 0 or 1.
 Publication:

arXiv eprints
 Pub Date:
 February 2009
 arXiv:
 arXiv:0902.4011
 Bibcode:
 2009arXiv0902.4011S
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 05A05 (Primary) 06A07;
 37F20 (Secondary)
 EPrint:
 final version