arXiv subjects

Timothy Y. Chow

Publications and source records attributed to Timothy Y. Chow.

23 records · Page 2Linked to original sources

arXiv2002

A note on a combinatorial interpretation of the e-coefficients of the chromatic symmetric function

Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting byproduct we derive a previously unnoticed result relating acyclic orientations to P-tableaux.

math.CO↗

arXiv2001

Symmetric function generalizations of graph polynomials

In Chapter 2 we study the path-cycle symmetric function of a digraph, a symmetric function generalization of Chung and Graham's cover polynomial. Most of this material appears in either Advances in Math. 118 (1996), 71-98 or J. Algebraic Combin. 10 (1999), 227-240. Chapter 3 contains miscellaneous results about Stanley's symmetric function generalization X_G of the chromatic polynomial, e.g., we establish a connection with some of Tutte's work on the chromatic polynomial and use this to prove that X_G is reconstructible. Most of Chapter 3 does not appear elsewhere.

math.CO↗

arXiv1998

What is a closed-form number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.

math.NT↗

arXiv1998

An independent set axiomatization for symplectic matroids

One of the simplest axiomatizations of a matroid is in terms of independent sets. Curiously, no such independent set axiomatization is known for Gelfand and Serganova's WP-matroids (which are ``Coxeter group analogues'' of matroids). Here we state and prove such an axiomatization in the special case of symplectic matroids. No prior knowledge of WP-matroids is assumed; we believe the paper should be accessible and interesting to anyone with some interest in matroids.

math.CO↗

arXiv1997

Descents, quasi-symmetric functions, and the chromatic symmetric function

We give a new proof of Chung and Graham's ``G-descent expansion'' of the classical chromatic polynomial, as well as a special case of the quasi-symmetric function expansion of the path-cycle symmetric function Xi_D. Both proofs rely on Stanley's quasi-symmetric function expansion of the chromatic symmetric function X_G. We also show that Stanley's expansion suggests that a Robinson-Schensted algorithm for (3+1)-free posets---something that has been sought for unsuccessfully for some time---ought to ``respect descents'' in a certain precise sense.

math.CO↗