We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously. In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.

We initiate a systematic study of key-avoidance on alternating sign matrices (ASMs) defined via pattern-avoidance on an associated permutation called the \emph{key} of an ASM. We enumerate alternating sign matrices whose key avoids a given set of permutation patterns in several instances. We show that ASMs whose key avoids $231$ are permutations, thus any known enumeration for a set of permutation patterns including $231$ extends to ASMs. We furthermore enumerate by the Catalan numbers ASMs whose key avoids both $312$ and $321$. We also show ASMs whose key avoids $312$ are in bijection with the gapless monotone triangles of [Ayyer, Cori, Gouyou-Beauchamps 2011]. Thus key-avoidance generalizes the notion of $312$-avoidance studied there. Finally, we enumerate ASMs with a given key avoiding $312$ and $321$ using a connection to Schubert polynomials, thereby deriving an interesting Catalan identity.

We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and generalizations, some of which we apply here. We give the classification of several classes of sortable inversion sequences in terms of pattern avoidance. We are able to provide an exact enumeration of some of the sortable classes in question using both classical approaches and a more recent strategy utilizing generating trees.

We consider sorting procedures for permutations making use of pop stacks with a bypass operation, and explore the combinatorial properties of the associated algorithms.

We define four new shuffling algorithms on permutations. For each algorithm, we characterize and enumerate the sets of permutations that are sorted after k iterations of the algorithm and we determine properties of the corresponding generating functions. For three of the algorithms, the sets of sortable permutations can be seen to be permutation classes for any nonnegative integer k.

We study permutations p such that both p and p2 avoid a given pattern q. We obtain a generating function for the case of q=312 (equivalently, q=231), we prove that if q is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for q=321. We also present some intriguing questions in the case of q=132.

We consider the set of permutations that are sortable after two passes through a pop stack. We characterize these permutations in terms of forbidden patterns and enumerate them according to the ascent statistic. Then we show these permutations to be in bijection with a special family of polyominoes.

We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of $(t+1)$-reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length $n$ was discovered along with a bijection between this family and the collection of alternating permutations of length $n-1$. We calculate generating functions for the number permutations of length $n$ and exact rev-tier $t$.

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

The vast and prolific area of permutation patterns considers permutations as linear orders. In this concept, a permutation p is a linear array p1p2 · · · pn of the first n positive integers in some order, so that each integer occurs exactly once. We say that a permutation p contains the pattern q = q1q2 · · · qk if there is a k-element set of indices i1 < i2 < · · · < ik so that pir < pis if and only if qr < qs. If p does not contain q, then we say that p avoids q. A recent survey on permutation patterns can be found in [9] and a book on the subject is [4]. This definition does not consider the other perspective from which permutations can be studied, namely that of the symmetric group, where the product of two permutations is defined, and the notion of a permutation’s inverse is defined. Therefore, it is not surprising that pattern avoidance questions become much more difficult if the symmetric group concept is present in them. (See [1], [2] or [5] for a few results in this direction.) One exception to this is the straightforward observation [3] that if p avoids q, then its inverse permutation p−1 avoids q−1.

In this paper, we study the generating functions for the number of set partitions of according to the number/sum of elements protected by weak/strong records (smaller elements that follow the weak/strong records). In particular, we present explicit formulas and asymptotics for the total number of number/sum of elements protected by weak/strong records over set partitions of [n].

Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the Robinson-Schensted correspondence. As a byproduct, we get an interesting generalization of layered permutations.

We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schröder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.

We study sorting machines consisting of a stack and a pop stack in series, with or without a queue between them. While there are, a priori, four such machines, only two are essentially different: a pop stack followed directly by a stack, and a pop stack followed by a queue and then by a stack. In the former case, we obtain complete answers for the basis and enumeration of the sortable permutations. In the latter case, we present several conjectures.

We study a sorting machine consisting of two stacks in series where the first stack has the added restriction such that entries in the stack must be in decreasing order from top to bottom. We give the basis of the class of permutations that are sortable by this machine which shows that it is enumerated by the Schröder numbers.

We study a sorting machine consisting of two stacks in series where the first stack has the added restriction such that entries in the stack must be in decreasing order from top to bottom. We give the basis of the class of permutations that are sortable by this machine which shows that it is enumerated by the Schröder numbers.

The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set of explicit families.

Masters Research - Masters of Philosophy (Mathematics)%%%%This thesis aims to provide an overview of the current works in the field of fractal sets produced via iterative mapping systems on circles. We will review the work of Frame, Mandelbrot and Neger(2005) in relation to circle inversion limit sets in the extended real plane. For the purpose of this thesis we will restrict our review to circle placements which are non-overlapping. Two circles will be classified non-overlapping if they have disjoint interiors and at most one point of contact between their boundaries. We will see how circle inversion limit sets can be created via an iterative system of non-expansive circle inversion mappings on non-overlapping circle placements, termed restricted iterated circle inversion systems (restricted ICISs). The work of Frame et al (2005) will be extended by developing a classification theorem for n non-overlapping circles in the extended real plane. This theorem predicts the structure of a given restricted ICIS limit set based on the placement of the initial inversion circles. We will see how the properties of circle inversion and circle inversion iteration can be extended to three dimension in the form of spherical inversion/ spherical inversion iteration. Once again we will limit ourselves to iterative systems using only non-expansive inversion mappings, termed restricted iterated spherical inversion systems (restricted ISISs). Limit sets formed under restricted ISISs on non-overlapping spherical placements will also be discussed. This thesis also explores iteration of mappings applied to circles in the extended complex plane. We will begin by reviewing the work of Mumford, Series and Wright (2002). Mumford et al (2002) formed fractal limit set by iterating systems of circle pairing Mobius transformations on specific non-overlapping circle placements. We will review the criteria needed for fractal limit sets to be formed under such an iterative system. Iteration of systems of complex circle inversion mappings will also be discussed as well as the structures of possible limit set. We will conclude this thesis with possible avenues for further work within the field of Fractal Prducing Iterative Mapping Systems on Circles.

Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.