Abstract
We investigate permutations and involutions that avoid a pattern of length three and have a unique longest increasing subsequence (ULIS). We prove an explicit formula for 231-avoiders, we show that the growth rate for 321-avoiding permutations with a ULIS is 4, and prove that their generating function is not rational. We relate the case of 132-avoiders to the existing literature, raising some interesting questions. For involutions, we construct a bijection between 132-avoiding involutions with a ULIS and bidirectional ballot sequences.
Highlights
Let p = p1p2 · · · pn be a permutation
We say that p has a unique longest increasing subsequence, or ULIS, if p has an increasing subsequence that is longer than all other increasing subsequences
The diverse nature of the results we prove will be interesting as we will see that depending on the pattern q, the portion of q-avoiding permutations that have a ULIS may converge to a positive constant, converge to 0 at a subexponential speed, or converge to zero at an exponential speed
Summary
Finding the number of all permutations of length n that have a unique longest increasing subsequence appears to be a difficult problem. These numbers are known only for n 15, given by sequence A167995 in the Online Encyclopedia of Integer Sequences [9]. We instead consider permutations that avoid a given pattern q of length three that have a ULIS. If q is any given pattern of length three, it is well known that the number of all permutations of length n that avoid q is the Catalan number Cn =. We denote by in(q) the number of q-avoiding involutions of length n which have a ULIS.
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