Finding and Counting Permutations via CSPs

Abstract Permutation patterns and pattern avoidance have been intensively studied in combinatorics computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps most natural algorithmic question this area is deciding whether a given permutation length n contains k . In we give two new algorithms for well-studied problem, one whose running time $$n^{k/4 + o(k)}$$ nk/4+o(k) , polynomial-space algorithm better $$O(1.6181^n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(1.6181n) $$O(n^{k/2 1})$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(nk/2+1) These results improve earlier best bounds $$n^{0.47k xmlns:mml="http://www.w3.org/1998/Math/MathML">n0.47k+o(k) $$O(1.79^n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(1.79n) due Ahal Rabinovich (2000) resp. Bruner Lackner (2012) are fastest problem when $$k \in \varOmega (\log {n})$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">k??(logn) We show that both our previous exponential-time literature can be viewed through unifying lens constraint-satisfaction Our also count within same time, number occurrences pattern. result close optimal: solving counting $$f(k) \cdot n^{o(k/\log {k})}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">f(k)·no(k/logk) would contradict hypothesis (ETH). For some special classes obtain improved times. further prove 3- increasing (4321-avoiding) decreasing (1234-avoiding) permutations can, sense, embed arbitrary almost linear length, which indicates sub-exponential unlikely with current techniques, even from these restricted classes.