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We review and extend what is known about the generating functions forconsecutive pattern-avoiding permutations of length 4, 5 and beyond, and theirasymptotic behaviour. There are respectively, seven length-4 and twenty-fivelength-5 consecutive-Wilf classes. D-finite differential equations are knownfor the reciprocal of the exponential generating functions for four of thelength-4 and eight of the length-5 classes. We give the solutions of some ofthese ODEs. An unsolved functional equation is known for one more class oflength-4, length-5 and beyond. We give the solution of this functionalequation, and use it to show that the solution is not D-finite. For threefurther length-5 c-Wilf classes we give recurrences for two and adifferential-functional equation for a third. For a fourth class we find a newalgebraic solution. We give a polynomial-time algorithm to generate thecoefficients of the generating functions which is faster than existingalgorithms, and use this to (a) calculate the asymptotics for all classes oflength 4 and length 5 to significantly greater precision than previously, and(b) use these extended series to search, unsuccessfully, for D-finite solutionsfor the unsolved classes, leading us to conjecture that the solutions are notD-finite. We have also searched, unsuccessfully, for differentially algebraicsolutions.

On consecutive pattern-avoiding permutations of length 4, 5 and beyond