Published online by Cambridge University Press: 05 October 2010
We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions.
Monotone subsequences in a permutation p = p1p2 … pn have been the subject of vigorous research for over sixty years. In this paper, we will review three different lines of work. In all of them, we will consider increasing subsequences of a permutation of length n that have a fixed length k. This is in contrast to another line of work, started by Ulam more than sixty years ago, in which the distribution of the longest increasing subsequence of a random permutation has been studied. That direction of research has recently reached a high point in the article of Baik, Deift, and Johansson.
The three directions we consider are distinguished by their definition of monotone subsequences. We can simply require that k entries of a permutation increase from left to right, or we can in addition require that these k entries be in consecutive positions, or we can even require that they be consecutive integers and be in consecutive positions.
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