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Séminaire Lotharingien de Combinatoire, B68a (2012), 20 pp.

# Sergey Kitaev and Jeffrey Remmel

# Quadrant Marked Mesh Patterns in Alternating Permutations

**Abstract.**
This paper is a continuation of the systematic study of the distribution of
quadrant marked mesh patterns initiated in
[*J. Integer Sequences*, **12** (2012), Article 12.4.7].
We study quadrant marked mesh patterns on
up-down and down-up permutations, also known as alternating and
reverse alternating permutations, respectively. In particular, we
refine classical enumeration results of André
[*C. R. Acad. Sci. Paris* **88** (1879), 965-967;
*J. Math. Pur. Appl.* **7** (1881), 167-184]
on alternating permutations by showing that the distribution
with respect to the
quadrant marked mesh pattern of interest is given by
(sec(*xt*))^{1/x} on up-down permutations of even length and by
on down-up permutations of odd
length.

Received: May 2, 2012.
Accepted: September 25, 2012.
Final Version: September 28, 2012.

The following versions are available:

## Corrigendum

On page 8, line 4 from the bottom, in the summation index
*UD*_{2n}^{(2k)} should be replaced by
*DU*_{2n}^{(2k)}.
On page 9, line 2, in the summation index
*UD*_{2n}^{(2k)} should be replaced by
*DU*_{2n}^{(2k+1)}.

On page 13, first line after (3.2),
*A*_{2k+2}^{(k+1)+k)} should be replaced by
*A*_{2k+2}^{=((k+1)+k)}.

On page 19, line 10 above References, the definition of (*x*)_{n}
should be corrected to

(*x*)_{n}=*x*(*x*+1)...(*x*+*n*-1)