Jeffrey Liese^{1} and Jeffrey Remmel^{2}

^{1}Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA
93407, USA

jliese@calpoly.edu

^{2}Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112,
USA

remmel@math.ucsd.edu

Annals of Combinatorics 14 (1) pp.123-141 Springer, 2010

**AMS Subject Classification: **05A05, 05A15

**
Abstract: **

Given a permutation τ of length j, we say that a permutation σ has a τ-match
starting at position i, if the elements in positions i, i+1, …, i+ j-1 in s have the same relative
order as the elements of τ. We have been able to take advantage of the results of Mendes and
Remmel [1] to obtain a generating function for the number of permutations with no τ-matches
for several new classes of permutations. These new classes include a large class of permutations
which are shuffies of an increasing sequence 1 2…n with an arbitrary permutation s of the
integers {n + 1, …, n +m}. Finally we give a formula for the generating function for the
number of permutations which have no 1 3 2 4-matches.

**Keywords: **permutation patterns, consecutive patterns, generating functions

**
References:**

1. Mendes, A., Remmel, J.: Permutations and words counted by consecutive patterns. Adv.
Appl. Math. 37(4), 443--480 (2006)

2. Elizalde, S., Noy, M.: Consecutive patterns in permutations. Adv. Appl. Math. 30(1-2),
110--125 (2003)

3. Kitaev, S.: Generalized patterns in words and permutations. Ph.D. Thesis, Chalmers University
of Technology and Göteborg University, Göteborg (2003)

4. Kitaev, S.: Partially ordered generalized patterns. Discrete Math. 298, 212--229 (2005)

5. Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. John Wiley & Sons Inc, New
York (1983)

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