Pattern frequency sequences and internal zeros

Authors:
Miklós Bóna;Bruce E. Sagan;Vincent R. Vatter
Affiliations:
Department of Mathematics, University of Florida, Gainesville, Florida;Department of Mathematics, Michigan State University, East Lansing, Michigan;Department of Mathematics, Rutgers University, Piscataway, New Jersey
Venue:
Advances in Applied Mathematics - Special issue: Memory of Rodica Simon
Year:
2002

Citing 6
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Abstract

Let q be a pattern and let Sn,q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (Sn,q(c))c≥0 has internal zeros. If q is a monotone pattern it turns out that, except for q = 12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q = 1(l + 1)l... 2 there are infinitely many sequences which contain internal zeros and when l = 2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.