2026/07/23 by Robert Laudone
#math.CO
We construct a new reduction process which takes a (132,213)-avoiding permutation to a shorter one that is cyclic if and only if the original was. Iterating it determines whether a given (132,213)-avoiding permutation is cyclic. Reversing it gives four moves that build every cyclic (132,213)-avoiding permutation, uniquely, from 1 if n is odd, and 21 if n is even. Our main application is the first non-trivial lower bound for the growth rate of Cn(132,213), the cyclic permutations of length n avoiding 132 and 213. We also give several other consequences of the reduction, including a bijection between the odd and even size classes and an exact enumeration for those permutations with a restricted number of layers.