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Séminaire Lotharingien de Combinatoire, B46g (2001), 12 pp.

# Toufik Mansour

#
Pattern Avoidance in Coloured Permutations

**Abstract.**
Let *S*_{n} be the symmetric group,
*C*_{r} the cyclic group of order *r*, and
let *S*_{n}^{(r)} be the wreath product of
*S*_{n} and *C*_{r};
which is the set of all coloured permutations on the symbols 1,2,...,*n*
with colours 1,2,...,*r*, which is the analogous of the symmetric group
when *r*=1,
and the hyperoctahedral group when *r*=2.
We prove, for every 2-letter coloured pattern *\phi* in
*S*_{2}^{(r)}, that
the
number of *\phi*-avoiding coloured permutations in
*S*_{n}^{(r)} is given by
the
formula *\sum_{j=0}^n j! (r-1)^j {\binom n j}^2*. Also we prove that the
number of Wilf classes of restricted coloured permutations by two patterns
with *r* colours in
*S*_{2}^{(r)} is one for *r*=1,
is four for *r*=2, and
is six for *r*>=3.

Received: June 24, 2001; Accepted: Oct. 17, 2001.

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