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1	Listing Combinatorial Objects January 2012 (has links) abstract: Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation. / Dissertation/Thesis / Ph.D. Mathematics 2012 Mathematics Gray codes overlap cycles universal cycles
2	Universal and Overlap Cycles for Posets, Words, and Juggling Patterns King, Adam, Laubmeier, Amanda, Orans, Kai, Godbole, Anant 01 May 2016 (has links) We discuss results dealing with universal cycles (ucycles) and s-overlap cycles, and contribute to the body of those results by proving existence of universal cycles of naturally labeled posets (NL posets), s-overlap cycles of words of weight k, and juggling patterns. The result on posets is, to the best of our knowledge, the first demonstration of the existence of a ucycle whose length is unknown. arc digraph labeled posets universal cycles Mathematics and Statistics
3	Universal and Near-Universal Cycles of Set Partitions Higgins, Zach, Kelley, Elizabeth, Sieben, Bertilla, Godbole, Anant 23 December 2015 (has links) We study universal cycles of the set P(n,k) of k-partitions of the set [n]:={1,2,…,n} and prove that the transition digraph associated with P(n,k) is Eulerian. But this does not imply that universal cycles (or ucycles) exist, since vertices represent equivalence classes of partitions. We use this result to prove, however, that ucycles of P(n,k) exist for all n≥3 when k=2. We reprove that they exist for odd n when k=n−1 and that they do not exist for even n when k=n−1. An infinite family of (n,k) for which ucycles do not exist is shown to be those pairs for which (Formula presented) is odd (3≤k coverings packings set partitions universal cycles Mathematics and Statistics
4	Shift gray codes Williams, Aaron Michael 11 December 2009 (has links) Combinatorial objects can be represented by strings, such as 21534 for the permutation (1 2) (3 5 4), or 110100 for the binary tree corresponding to the balanced parentheses (()()). Given a string s = s1 s2 sn, the right-shift operation shift(s, i, j) replaces the substring si si+1..sj by si+1..sj si. In other words, si is right-shifted into position j by applying the permutation (j j−1 .. i) to the indices of s. Right-shifts include prefix-shifts (i = 1) and adjacent-transpositions (j = i+1). A fixed-content language is a set of strings that contain the same multiset of symbols. Given a fixed-content language, a shift Gray code is a list of its strings where consecutive strings differ by a shift. This thesis asks if shift Gray codes exist for a variety of combinatorial objects. This abstract question leads to a number of practical answers. The first prefix-shift Gray code for multiset permutations is discovered, and it provides the first algorithm for generating multiset permutations in O(1)-time while using O(1) additional variables. Applications of these results include more efficient exhaustive solutions to stacker-crane problems, which are natural NP-complete traveling salesman variants. This thesis also produces the fastest algorithm for generating balanced parentheses in an array, and the first minimal-change order for fixed-content necklaces and Lyndon words. These results are consequences of the following theorem: Every bubble language has a right-shift Gray code. Bubble languages are fixed-content languages that are closed under certain adjacent-transpositions. These languages generalize classic combinatorial objects: k-ary trees, ordered trees with fixed branching sequences, unit interval graphs, restricted Schr oder and Motzkin paths, linear-extensions of B-posets, and their unions, intersections, and quotients. Each Gray code is circular and is obtained from a new variation of lexicographic order known as cool-lex order. Gray codes using only shift(s, 1, n) and shift(s, 1, n−1) are also found for multiset permutations. A universal cycle that omits the last (redundant) symbol from each permutation is obtained by recording the first symbol of each permutation in this Gray code. As a special case, these shorthand universal cycles provide a new fixed-density analogue to de Bruijn cycles, and the first universal cycle for the "middle levels" (binary strings of length 2k + 1 with sum k or k + 1). shorthand universal cycles combinatorial generation minimal-change order loopless algorithm efficient algorithm combinations multiset permutations balanced parentheses Dyck words Catalan paths Schroder paths Motzkin words linear-extensions posets connected unit interval graphs inversions binary trees k-ary trees Lyndon words pre-necklaces theoretical computer science discrete mathematics combinatorics brute forcs de Bruijn cycles bubble languages cool-lex order lexicographic order combinatorial enumeration stacker-crane problem traveling salesman problem middle levels fixed-density de Bruijn cycle fixed-content

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