Avoidance of vincular patterns by flattened derangements
Toufik Mansour, Mark Shattuck
TL;DR
This work studies the avoidance of vincular patterns of length three by flattened-derangements and derives explicit generating-function descriptions for the corresponding avoidance classes according to the number of cycles. The authors employ kernel-method techniques to solve functional equations arising from careful refinements of the flattened-form structure, with a special exponential-generating-function treatment for $23-1$ that leads to formulas involving Stirling numbers and tridiagonal determinants. For patterns of type $(2,1)$ and $(1,2)$, they provide detailed recurrences and generating-function frameworks, including exact cycle-distribution results for several cases and, in particular, demonstrations that certain patterns reduce to classical-pattern counts or yield closed forms in terms of Bell numbers. These results extend prior flat-pattern-avoidance work and connect derangement-pattern avoidance to determinant structures and classical combinatorial sequences, enriching the understanding of how cycle structure interacts with pattern avoidance under flattening.
Abstract
In this paper, we consider the problem of avoiding a single vincular pattern of length three by derangements in the flattened sense and find explicit formulas for the generating functions enumerating members of each corresponding avoidance class according to the number of cycles. We make frequent use of the kernel method in solving the functional equations that arise which are satisfied by these (ordinary) generating functions. In the case of avoiding 23-1, which is equivalent to 32-1 in the flattened sense, it is more convenient to consider the exponential generating function instead due to the form of the recurrence. This leads to an explicit expression for the distribution of the number of cycles in terms of Stirling numbers of the second kind and the determinant of a certain tridiagonal matrix. Finally, the cases of 3-12 and 3-21 are perhaps the most difficult of all, and here we make use of a pair of auxiliary statistics in order to find a system of recurrences that enumerate each avoidance class.