Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.1 |

Department of Mathematics

University of Washington

Seattle, WA 98195-4350

USA

Bruce E. Sagan

Department of Mathematics

Michigan State University

East Lansing, MI 48824-1027
USA

**Abstract:**

Let
denote the symmetric group of all permutations
of
.
An index *i* is a
*peak* of
if
*a*_{i-1}<*a*_{i}>*a*_{i+1} and we let
be
the set of peaks of .
Given any set *S* of positive integers we
define
.
Our main
result is that for all fixed subsets of positive integers *S*
and all sufficiently large *n* we have
for
some polynomial *p*(*n*) depending on *S*. We explicitly compute *p*(*n*)
for various *S* of probabilistic interest, including certain
cases where *S* depends on *n*. We also discuss two conjectures, one
about positivity of the coefficients of the expansion of *p*(*n*) in a
binomial coefficient basis, and the other about sets *S* maximizing
when
is fixed.

(Concerned with sequences A000431 A008303.)

Received December 3 2012;
revised version received May 22 2013.
Published in *Journal of Integer Sequences*, June 5 2013.

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