Pattern frequency sequences and internal zeros
With Miklós Bóna and Bruce Sagan.
Advances in Applied Mathematics 28 (2002), 395-420. doi:10.1006/aama.2001.0789.
Consider the number of permutations in the symmetric group on n letters that contain c copies of a given pattern. As c varies (with n held fixed) these numbers form a sequence whose properties we study for the monotone patterns and the patterns 1l(l-1)...2. We show that, except for the patterns 12 and 21 where the sequence is well-known to be log concave, there are infinitely many n where the sequence has internal zeros.
From the announcement for a talk based on this paper: “Warning: this talk will consist of a LOT of algebraic derivations.”
Download the paper:
- download from the journal website (subscription required)
- ps
- source
Related links:
- Bounding quantities related to the packing density of 1(l+1)l...2 builds upon this work.
- This work is briefly covered in the notes for a talk I gave at the Rutgers Graduate Student Combinatorics Seminar, which are available in pdf, ps, or tex.
- The sequence Mn for the pattern 132 is number A061061 in the OEIS.
- The sequence Mn for the pattern 1432 is number A100354 in the OEIS.
- The sequence Mn for the pattern 15432 is number A100355 in the OEIS.
- The sequence Mn for the pattern 165432 is number A100356 in the OEIS.