William Y. C. Chen

Parameter Augmentation for Basic Hypergeometric Series, II

In a previous paper, we explored the idea of parameter augmentation for basic hypergeometric series, which provides a method of provingq-summation and integral formula based special cases obtained by reducing some parameters to zero. In the present paper, we shall mainly deal with parameter augmentation forq-integrals such as the Askey?Wilson integral, the Nassrallah?Rahman integral, theq-integral form of Sears transformation, and Gaspers formula of the extension of the Askey?Roy integral. The parameter augmentation is realized by another operator, which leads to considerable simplications of some well knownq-summation and transformation formulas. A brief treatment of the Rogers?Szego polynomials is also given.

Cycle prefix digraphs for symmetric interconnection networks

Motivated by the study of large graphs with given degree and diameter, and the recent interest in the design of highly symmetric interconnection networks (e.g., the study of Cayley digraphs), we are led to the search for large vertex symmetric digraphs with given degree and diameter. The main result of this paper is the construction of a new class of vertex symmetric directed graphs, Γδ(D) (δ ≥ D) that have degree δ, diameter D, and (δ + 1)δ … (δ – D + 2) vertices. The graphs Γδ(D) are first found in the notation of Cayley coset digraphs. Then, we discover that they have a very simple representation in terms of sequences like the commonly studied networks such as the hypercube, de Bruijn graphs, and Kautz graphs. Based on the sequence representation, we give a simple shortest-path routing scheme. We also show that the average distance in our digraph Γδ(D) is very close to its diameter D. As a consequence, it follows that the natural routing scheme, which is even simpler than the shortest-path routing, is nearly optimal on an average basis.

The combinatorial power of the companion matrix

Abstract Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Warings formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials including two recently discovered classes that satisfy analogous smallest-norm and orthogonality properties subject to different weight functions, Dickson polynomials of various kinds arising from the theory of finite fields, combinatorial expansions of Toeplitz matrices, and the recent notion of cycle dissections involving a bijective study of Warings formula.

The factorial Schur function

The application of the divided difference of a function to the inhomogeneous symmetric functions (factorial Schur functions) of Biedenharn and Louck is shown to lead to new relations and simplified proofs of their properties. These results include determinantal definitions and the factorial Jacobi–Trudi identities with extensions to skew versions. Similar properties of a second class of symmetric functions depending on an arbitrary parameter, and of importance for generalized hypergeometric functions and series, are shown also to be derivable from the divided difference notion, slightly extended.

Context-free grammars, differential operators and formal power series

Abstract In this paper, we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules. We call such a set of rules a context-free grammar because these rules act like a context-free grammar in the sense of a formal language. Given a context-free grammar, we can associate each formal function with an exponential formal power series. In this way, we obtain grammatical interpretations of addition, multiplication and functional composition of formal power series. A surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation. We apply this method to obtain simple demonstrations of Faa di Brunos formula, and some identities concerning Bell polynomials, Stirling numbers and symmetric functions. In particular, the Lagrange inversion formula has a simple grammatical representation. From this point of view, one sees that Cayleys formula on labeled trees is equivalent to the Lagrange inversion formula.

q -Analogs of the inclusion-exclusion principle and permutations with restricted position

Abstract We derive a q -analog of the principle of inclusion-exclusion, and use it to derive a q -analog of the Kaplansky–Riordan theory of permutations with restricted position. Some analogies with the theory of Mahonian statistics are pointed out at the end, leading to a conjectured relationship between the two.

Random k-noncrossing RNA structures

In this paper, we introduce a combinatorial framework that provides an interpretation of RNA pseudoknot structures as sampling paths of a Markov process. Our results facilitate a variety of applications ranging from the energy-based sampling of pseudoknot structures as well as the ab initio folding via hidden Markov models. Our main result is an algorithm that generates RNA pseudoknot structures with uniform probability. This algorithm serves as a steppingstone to sequence-specific as well as energy-based transition probabilities. The approach employs a correspondence between pseudoknot structures, parametrized in terms of the maximal number of mutually crossing arcs and certain tableau sequences. The latter can be viewed as lattice paths. The main idea of this paper is to view each such lattice path as a sampling path of a stochastic process and to make use of D-finiteness for the efficient computation of the corresponding transition probabilities.

The theory of compositionals

Abstract By introducing the notion of compositionals we obtain a combinatorial interpretation of plethysm of formal power series in infinitely many variables. The following related problems are studied: Polyas theorem on the plethysm of cycle indices, plethystic inverse, the inverse of a sequence of delta series, the plethystic analog of the partition lattice, the reduced incidence algebra of the plethystic lattice, and the plethystic Hopf algebra. We also introduce plethystic trees, enriched plethystic trees and plethystic Schroder trees. An involution for plethystic Schroder trees is devised, which leads to a combinatorial expansion for the plethystic inverse of a series f ( x 1 , x 2 ,…) containing the factor x 1 .

The skew, relative, and classical derangements

Abstract In his popular combinatories text, Brualdi elucidates the principle of inclusion and exclusion with the classical and the relative derangements. Eventually, the two kinds of derangements are linked up via an algebraic relationship from the parallel use of the principle of inclusion and exclusion. We introduce the notion of skew derangements and relate them to relative derangements and the classical derangements by a purely combinatorial correspondence. Moreover, with the aid of our bijection we easily generalize the relative derangements, obtaining a binomial-type formula for the number of such generalized relative derangements on n elements in terms of the classical derangement number.

The Pessimistic Search and the Straightening Involution for Trees

We introduce the idea of pessimistic search on a rooted tree, and develop the straightening involution to relate the inversion polynomial evaluated atq=?1 to the number of even rooted trees. We obtain a differential equation for the inversion polynomial of cyclic trees evaluated atq=?1, a problem proposed by Gessel, Sagan and Yeh. Some brief discussions about relevant topics are also presented.