On the Combinatorics of Lecture Hall Partitions

A lecture hall partition of length n is an integer sequence satisfying Bousquet-Mélou and Eriksson showed that the number of lecture hall partitions of length n of a positive integer N whose alternating sum is k equals the number of partitions of N into k odd parts less than 2n. We prove the fact by a natural combinatorial bijection. This bijection, though defined differently, is essentially the same as one of the bijections found by Bousquet-Mélou and Eriksson.     

Characterization of reaction products of iron and iron salts and aqueous plant extracts

The complexes formed in aqueous solution as a result of a reaction of iron and iron salts (Fe²⁺ and Fe³⁺) and some plant extracts were analyzed using Mössbauer spectroscopy and Fourier transform infrared. The extracts were obtained from Opuntia elatior mill., Acanthocereus pentagonus (L.) Britton, Mimosa tenuiflora, Caesalpinia coriaria (Jacq.) Willd., Bumbacopsis quinata (Jacq.) Dugand and Acacia mangium Willd., plants growing wildly in different zones of the Isthmus of Panama. Results suggest the formation of mono- and bis-type complexes, and in some cases, the occurrence of a redox reaction. The feasibility of application of the studied extracts as atmospheric corrosion inhibitors is discussed.     

Manganese Dioxide Mediated Stereoselective, One-step Synthesis of Novel Steroidal (6R)-Spiro-1’,2’,4’-triazoline-3’-thiones

Oxidative cyclization of 5α-cholestan-6-one thiosemicarbazone (1), its 3β-chloro (2) and 3β-acetoxy (3) analogues with active manganese dioxide at room temperature afforded selectively the corresponding (6R)-spiro-1’,2’,4’-triazoline-3’-thiones 4—6. This synthesis has the advantages of mild reaction conditions, easy handling, easily available reagent and high yields. The products have been characterized by analytical and spectral data.     

Morphological,thermal, and conductivity studies of poly(methyl methacrylate)/polyaniline dodecylbenzenesulfonate blends

Soluble in organic solvent polyaniline (PANI) was prepared using DBSA as a functional dopant. Miscibility was maximized for PMMA with hydroquinone. Transmission electron micrographs showed lowest level of phase separation. Thermal studies by differential scanning calorimetry show some peak shifts which indicate the miscibility of two polymers. The formations of new bonds are also confirmed by Fourier transformation infrared spectroscopy. Electroconductivity increased with increase of content of doped PAN1, showing percolation threshold at 3 wt %.     

Synthesis of Pd-CNT nanocomposites and investigation of their catalytic behavior in the hydrodehalogenation of aryl halides

A catalytic system based on Pd-CNT nanocomposites for the hydrodehalogenation of aryl halides is described. Thiol groups were utilized as linkers to secure the Pd nanoparticles without agglomeration. The Pd-CNT nanocomposites effectively promoted the hydrodehalogenation of aryl halides at a low Pd content (∼2.3%) and in the absence of any ligand. The results suggest that the CNTs could significantly influence the catalytic activities of CNT-supported metal catalysts for hydrodehalogenation.     

Diethyl 1-(p-fluoro­phenyl)-5-oxo-3-(2-thienyl)­pyrrolidine-2,2-di­carboxyl­ate

In the title compound, C₂₀H₂₀FNO₅S, the pyrrolidine ring adopts an envelope conformation. The fluoro­phenyl and thio­phene rings are individually planar. The molecular and crystal structures are stabilized by intra- and intermolecular C—H⋯O interactions.     

On the Tannic Acid Interaction with Metallic Iron

The reaction between tannic acid and metallic iron was studied using infrared and Mössbauer spectroscopy. Iron converts to sparingly soluble and amorphous ferric tannate complexes. The degree of conversion was followed from 1 day to 6 months. In the very early stages of reaction the product consists of a mono-type complex, while in the later stages a mixture of mono- and bis-type complexes were formed. The conversion reaction of metallic iron to tannate complexes follows a first-order reaction kinetics.     

Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.