258 research outputs found
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
We say that a permutation is a Motzkin permutation if it avoids 132 and
there do not exist such that . We study the
distribution of several statistics in Motzkin permutations, including the
length of the longest increasing and decreasing subsequences and the number of
rises and descents. We also enumerate Motzkin permutations with additional
restrictions, and study the distribution of occurrences of fairly general
patterns in this class of permutations.Comment: 18 pages, 2 figure
Permutations with restricted patterns and Dyck paths
We exhibit a bijection between 132-avoiding permutations and Dyck paths.
Using this bijection, it is shown that all the recently discovered results on
generating functions for 132-avoiding permutations with a given number of
occurrences of the pattern follow directly from old results on the
enumeration of Motzkin paths, among which is a continued fraction result due to
Flajolet. As a bonus, we use these observations to derive further results and a
precise asymptotic estimate for the number of 132-avoiding permutations of
with exactly occurrences of the pattern . Second,
we exhibit a bijection between 123-avoiding permutations and Dyck paths. When
combined with a result of Roblet and Viennot, this bijection allows us to
express the generating function for 123-avoiding permutations with a given
number of occurrences of the pattern in form of a continued
fraction and to derive further results for these permutations.Comment: 17 pages, AmS-Te
321-polygon-avoiding permutations and Chebyshev polynomials
A 321-k-gon-avoiding permutation pi avoids 321 and the following four
patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1),
k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k,
(k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were
introduced and studied by Billey and Warrington [BW] as a class of elements of
the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the
singular loci of whose Schubert varieties have fairly simple formulas and
descriptions. Stankova and West [SW] gave an exact enumeration in terms of
linear recurrences with constant coefficients for the cases k=2,3,4. In this
paper, we extend these results by finding an explicit expression for the
generating function for the number of 321-k-gon-avoiding permutations on n
letters. The generating function is expressed via Chebyshev polynomials of the
second kind.Comment: 11 pages, 1 figur
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