S. Gill Williamson

A loop-free algorithm for generating the linear extensions of a poset

A precise concept of when a combinatorial counting problem is “hard” was first introduced by Valiant (1979) when he defined the notion of a #P-complete problem. Correspondingly, there has been consistent interest in the notion of when a combinatorial listing problem admits a very special regular structure in which transition times between objects being listed are uniformly bounded by a fixed constant. Early descriptions of suchloop-free listing algorithms may be found in the bookAlgorithmic Combinatorics by Even (1973). Recently, the problem of counting all linear extensions of a partially ordered set has received attention with regard to both of these combinatorial concepts. Brightwell and Winkler (1991) have shown, by a very ingenious argument, that the poset-extension counting problem is #P-complete. Pruesse and Ruskey (1992) have shown that the corresponding listing problem can be solved in constant amortized time and have posed the problem of finding a loop-free algorithm for the poset-extension problem. The present paper presents a solution to this latter problem. This sequence of results represents an interesting juxtaposition, in a fixed, naturally-occurring combinatorial problem, of intricate and precisely defined “irregularities” with respect to counting with very strong regularities with respect to listing.

A sequential sorting network analogous to the batcher merge

We present a network of delay log2 N, whose comparators have only log2 N different lengths with maximum length N/2. This network is log-sequential in that it will sort N data items when they are passed through it log2 Ntimes. The design, which is related to the Batcher odd-even merge, is distinctly different from the first known example of a log-delay log-sequential network, due to Dowd, Perl, Rudolf, and Saks. It is quite probably the best possible sorting network.

Ranking and unranking planar embeddings

In this paper, we give a formula for computing the number of different planat embeddings of any planar biconnected graph. The enumeration method used in deriving the formula readily gives rise to efficient algorithms for the ranking, unranking and random generation of embeddings of the given graph. We also give linear time algorithms for checking planarity and constructing any particular embedding.

Permanents and determinants with generic noncommuting entries

This paper studies some basic combinatorial properties of matrix functions of generic matrices. A generic matrix is one with entries from a free associative algebra, over a field, and on a finite set of non-commuting variables (i.e. a tensor algebra). The principal tools are shuffle products. Generic column and row permanents are defined and analogs of the Laplace and Cauchy-Binet theorems are derived in terms of shuffles. In this setting, the generic permanents include as special cases all of the classical matrix functions: Schur matrix functions, determinants, and permanents. 1980 Mathematics Classification 05, 15. Keywords: Shuffle product, generic matrix functions, minor expansions, Laplace Expansion Theorem, Cauchy-Binet Theorem, permanents, determinants, tensor algebra, matrices with non-commuting entries.

Canonical forms for cycles in bridge graphs

Let G = (V,E) be a biconnected graph and let C be a cycle in G. The subgraphs of G identified with the biconnected components of the contraction of C in G are called the bridges of C. Associated with the set of bridges of a cycle C is an auxilliary graphical structure GC called a bridge graph or an overlap graph. Such auxilliary graphs have provided important insights in classical graph theory, algorithmic graph theory, and complexity theory. In this paper, we use techniques from algorithmic combinatorics and complexity theory to derive canonical forms for cycles in bridge graphs. These canonical forms clarify the relationship between cycles in bridge graphs, the structure of the underlying graph G, and lexicographic order relations on the vertices of attachment of bridges of a cycle. The first canonical form deals with the structure of induced bridge graph cycles of length greater than three. Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characteri...

Generic common minor expansions

A classical variation on the Laplace expansion theorem relates the product of the determinant of a matrix and one of its fixed submatrices to an expansion by minors. This paper explores the extensions of this highly combinatorial result to generic matrix functions. The principal tool is the shuffle product. These ideas are used to extend the corresponding classical results for determinants and to study the case of permanents. An application is given to the problem of characterizing the actions (as differential operators) of exterior powers of the classical Cayley operators. The resulting identities involve hook length products of frames of integral partitions and generalize classical results of Cayley concerning the action of determinantal differential operators on polynomial functions of generic determinants.

A CLASS OF GRAPHS WHICH HAS EFFICIENT RANKING AND UNRANKING ALGORITHMS FOR SPANNING TREES AND FORESTS

Remmel and Williamson recently defined a class of directed graphs, called filtered digraphs, and described a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions [12]. Filtered digraphs include many specialized graphs such as complete k-partite graphs. The Remmel-Williamson bijections provide explicit formulas for various multivariate generating functions for the oriented spanning forests which arise in this context. In this paper, we prove another important property of these bijections, namely, that it allows one to construct efficient algorithms for ranking and unranking spanning trees or spanning forests of filtered digraphs G. For example, we show that if G=(V,E) is a filtered digraph and SP(G) is the collection of spanning trees of G, then our algorithm requires O(|V|) operations of sum, difference, product, quotient, and comparison of numbers less than or equal |SP(G)| to rank or unrank spanning trees of G.

Research problem: combinatorial and multilinear aspects of sign-balanced posets

We present basic results and open problems related to the study of sets L(P) of linear extensions of posets (P, ⪯). In our study, these linear extensions are regarded as permutations σ with the alternating character (-1)inv(σ). On the combinatorial side (Section 2), we present a number of open problems related to the fundamental study of sign-balanced posets whose Hasse diagrams are specified by Ferrers diagrams of partitions. On the multilinear-algebraic side (Section 3), we propose the general study of operators associated with P L(P), and the alternating character. We derive an antisymmetric decomposition of such operators as shuffles of anti-symmetric operators associated with symmetric groups. In general, we show that the set L(P)=G is a group if and only if P is an ordinal sum of antichains and G is a direct product of symmetric groups. As a consequence, the antisymmetric decomposition becomes a tensor product if and only if L(P) is a group.