Carla D. Savage

A Survey of Combinatorial Gray Codes

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

Fast, Efficient Parallel Algorithms for Some Graph Problems

Algorithms for solving graph problems on an unbounded parallel model of computation are considered. Parallel algorithms of time complexity

Monotone gray codes and the middle levels problem

O(\log ^2 n)

Generating necklaces

are described for finding biconnected components, bridges, minimum spanning trees and fundamental cycles. In the algorithms for finding minimum spanning trees, bridges, and fundamental cycles, the number of processors used is small enough that the parallel algorithm is efficient in comparison with the best sequential algorithms for these problems. Several other algorithms are presented which are especially suitable for processing sparse graphs.

The s -Eulerian polynomials have only real roots

Abstract An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of B n (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n, a hamilton path in B n with the following additional property: edges between levels i − 1 and i of B n must appear on the path before edges between levels i and i + 1. Two consequences are an embedding of the hypercube into a linear array which simultaneously minimizes dilation in both directions, and a long path in the middle two levels of B n. Using a second recursive construction, we are able to improve still further on this path, thus obtaining the best known results on the notorious “middle levels” problem (to show that the graph formed by the middle two levels of B 2k + 1 is hamiltonian for all k). We show in fact that for every ϵ > 0, there is an h ⩾ 1 so that if a hamilton cycle exists in the middle two levels of B 2k + 1 for 1 ⩽ k ⩽ h, then there is a cycle of length at least (1 − ϵ) N(k) for all k ⩾ 1, where N(k)=2(2kk+1). Using the fact that hamilton cycles are currently known to exist for 1 ⩽ k ⩽ 11, the construction guarantees a cycle of length at least 0.839N(k) in the middle two levels of B 2k + 1 for all k.

Hamilton cycles that extend transposition matchings in Cayley graphs of S n

A k color n bead necklace is an equivalence class of k ary n tuples under rotation In this paper we analyze an algorithm due to Fredricksen Kessler and Maiorana FKM to show that necklaces can be generated in constant amortized time We also present a new approach to generating necklaces which we conjecture can also be implemented in constant amortized time The FKM algorithm generates a list of n tuples which includes among other things the lexicographically smallest element of each k color n bead necklace Previously it had been shown only that the list contains at most O n N k n elements where N k n is the number of k color n bead necklaces and that successive elements can be generated in worst case time O n giving a bound of O n N k n on the time for the algorithm We show that the number of elements generated by the FKM algorithm approaches k k N k n and the total time is only O N k n A by product of our analysis is a precise characterization of the list generated by FKM which makes a recursive description possible

On the Multiplicity of Parts in a Random Composition of a Large Integer

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences s of positive integers, which they called s-inversion sequences. Our object of study is the generating polynomial of the ascent statistic over the set of s-inversion sequences of length n. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations, we call this generalized polynomial the s-Eulerian polynomial. The main result of this paper is that, for any sequence s of positive integers, the s-Eulerian polynomial has only real roots. This result is first shown to generalize several existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several q-analogs. We show that the MacMahon-Carlitz q-Eulerian polynomial has only real roots whenever q is a positive real number, confirming a conjecture of Chow and Gessel. The same holds true for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively. Our results have interesting geometric consequences as well.

Gray code sequences of partitions

Let B be a basis of transpositions for

A gray code for necklaces of fixed density

S_n

Anti-lecture hall compositions

and let Cay