Permutations with forbidden subsequences and a generalized Schröder number

AbstractUsing the technique of generating trees, we prove that there are exactly 10 classes of pattern avoiding permutations enumerated by the large Schröder numbers. For each integer, m⩾1, a sequence which generalizes the Schröder and Catalan numbers is shown to enumerate m+22 classes of pattern avoiding permutations. Combinatorial interpretations in terms of binary trees and polyominoes and a generating function for these sequences are given

Investigating First Returns: The Effect of Multicolored Vectors

By definition, a first return is the immediate moment that a path, using vectors in the Cartesian plane, touches the x-axis after leaving it previously from a given point; the initial point is often the origin. In this case, using certain diagonal and horizontal vectors while restricting the movements to the first quadrant will cause almost every first return to end at the point (2n,0), where 2n counts the equal number of up and down steps in a path. The exception will be explained further in the sections below. Using the first returns of Catalan, Schröder, and Motzkin numbers, which resulted from the lattice paths formed using a combination of diagonal and/or horizontal vectors, we then investigated the effect that coloring select vectors will have on each of the original generatin

Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and reference

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.Comment: 16 pages, 4 figure

Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations