A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

Feasible Graphs and Colorings

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring.     

Hybrid Answer Set Programming

This paper discusses an extension of Answer Set Programming (ASP) called Hybrid Answer Set Programming (H-ASP) which allows the user to reason about dynamical systems that exhibit both discrete and continuous aspects. The unique feature of Hybrid ASP is that it allows the use of ASP type rules as controls for when to apply algorithms to advance the system to the next position. That is, if the prerequisites of a rule are satisfied and the constraints of the rule are not violated, then the algorithm associated with the rule is invoked.     

A p, q-Analogue of the Generalized Derangement Numbers

In this paper, we study the numbers D _n,k which are defined as the number of permutations σ of the symmetric group S _n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D _n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D _n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D _n,1, denoted by D _n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D _n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D _n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2. Partially supported by NSF grant DMS 0400507.     

Analysis of metoprolol enantiomers in human serum by liquid chromatography on a cellulose-based chiral stationary phase

Metoprolol is a lipophilic, cardioselective beta-adrenergic blocking agent commercially available as a racemic compound. A normal phase high-performance liquid chromatographic method was developed to directly determine individual enantiomeric concentrations of metoprolol in human serum. Separation of the enantiomers was accomplished by a cellulose-tris(3,5-dimethylphenylcarbamate) chiral stationary phase. Metoprolol enantiomers were detected by means of fluorescence with excitation and emission wavelengths of 275 and 315 nm, respectively. Standard curves were linear over the concentration range 12.5-400 ng/ml for each enantiomer. Within-day coefficient of variation was less than 15% at all concentrations and the between-day coefficient of variation ranged from 4.1 to 11.2%. The limit of detection was determined to be 5 ng/ml for each enantiomer and the stereoselective resolution (alpha) of R- and S-metoprolol was 3.08. The assay was employed to determine enantiomeric serum concentrations of metoprolol in healthy male volunteers.     

Bijective proofs of formulae for the number of standard Yound tableaux

We give a bijective proof of a formula due independently to Frobenius and Young for the number of standard Young tableau of shape λ for λ any partition. Frame, Robinson, and Thrall derived their hook formula for the number of standard Young tableau from the Frobenius-Young formula. As a corollary to our bijective proof of the Frobenius-Young formula, we also give a bijective proof of the Frame-Robinson-Thrall hook formula.     

Liquid chromatographic assay of carbovir, a carbocyclic nucleoside active against human immunodeficiency virus

Carbovir is a novel carbocyclic guanosine derivative that has potent in vitro activity against human immunodeficiency virus, the causative agent of acquired immunodeficiency syndrome (AIDS). Two methods of sample preparation were developed for the analysis of carbovir in rat blood. Solid-phase extraction on C18 extraction columns proved to be the most effective. Whole rat blood (200 microliters) was diluted with 0.8 ml of distilled water containing the internal standard. After two freeze-thaw cycles to lyse the red blood cells and subsequent centrifugation at 13,000 g, the supernatant was loaded on the C18 extraction columns. Carbovir and the internal standard were eluted with methanol-water (60:40). The extract was evaporated and reconstituted in mobile phase and the samples were injected onto a high-capacity reversed-phase column. The compounds were detected at 252 nm. Other nucleosides that could be used in the treatment of AIDS such as zidovudine and acyclovir did not interfere. Standard curves were linear over the concentration range 0.156-28.0 micrograms/ml (r2 greater than 0.99). The within-day coefficient of variation was less than 7.6% at all concentrations (n = 4). The between-day coefficient of variation ranged from 16.7 to 2.0% (n = 14). The limit of sensitivity was 0.05 micrograms/ml with a 200-microliters blood sample and the average extraction recovery was 74%. Carbovir was stable in rat blood for at least 4 h at 37 degrees C. The assay was used to determine the blood levels of carbovir in a rat after a 20 mg/kg intravenous dose.