Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.6

## Analogues of Up-down Permutations for Colored Permutations

### Andrew Niedermaier and Jeffrey Remmel Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 USA

Abstract:

André proved that is the generating function of all up-down permutations of even length and is the generating function of all up-down permutation of odd length. There are three equivalent ways to define up-down permutations in the symmetric group . That is, a permutation in the symmetric group is an up-down permutation if either (i) the rise set of consists of all the odd numbers less than , (ii) the descent set of consists of all even number less than , or (iii) both (i) and (ii). We consider analogues of André's results for colored permutations of the form where and under the product order. That is, we define if and only if and . We then say a colored permutation is (I) an up-not up permutation if the rise set of consists of all the odd numbers less than , (II) a not down-down permutation if the descent set of consists of all the even numbers less than , (III) an up-down permutation if both (I) and (II) hold. For , conditions (I), (II), and (III) are pairwise distinct. We find -analogues of the generating functions for up-not up, not down-down, and up-down colored permutations.

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(Concerned with sequences A000111 A000182 A122045.)

Received December 19 2009; revised version received May 5 2010. Published in Journal of Integer Sequences, May 5 2010.

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