Finding and Counting Permutations via CSPs

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is n^{k/4 + o(k)}, and a polynomial-space algorithm whose running time is the better of O(1.6181^n) and O(n^{k/2 + 1}). These results improve the earlier best bounds of n^{0.47k + o(k)} and O(1.79^n) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k in Omega(log{n}). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f(k) * n^{o(k/log{k})} would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes

Meander Graphs

We consider a Markov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meander $M = [A:B]$ is formed by two noncrossing perfect matchings, above $A$ and below $B$ the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on $A$ and the other on $B$. We also prove that the subset of meanders with a fixed $B$ is connected under a suitable local move operating on an appropriately defined meandric triple in $A$. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open

Bounds for the Growth Rate of Meander Numbers

We provide improvements on the best currently known upper and lower bounds for the exponential growth rate of meanders. The method of proof for the upper bounds is to extend the Goulden-Jackson cluster method

Bounds for the growth rate of meander numbers

We provide improvements on the best currently known upper and lower bounds for the exponential growth rate of meanders. The method of proof for the upper bounds is to extend the Goulden-Jackson cluster method. (c) 2005 Elsevier Inc. All rights reserved