On consecutive patternavoiding permutations of length 4, 5 and beyond
Abstract
We review and extend what is known about the generating functions for consecutive patternavoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length4 and twentyfive length5 consecutiveWilf classes. Dfinite differential equations are known for the reciprocal of the exponential generating functions for four of the length4 and eight of the length5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length4, length5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not Dfinite. For three further length5 cWilf classes we give recurrences for two and a differentialfunctional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomialtime algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for Dfinite solutions for the unsolved classes, leading us to conjecture that the solutions are not Dfinite. We have also searched, unsuccessfully, for differentially algebraic solutions.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.08839
 Bibcode:
 2017arXiv170408839B
 Keywords:

 Mathematics  Combinatorics;
 05Exx;
 05Axx
 EPrint:
 23 pages, 2 figures (update of references, plus web link to enumeration data). Minor update. Typos corrected. One additional reference