On connection sequence for equivalent polynomial sets

The main purpose of this paper is to study the connection sequence between two polynomial sets. We prove that any two equivalent polynomial sets can be connected with a Sheffer sequence. The principal result presents a unification of some useful known results. The inversion, addition and duplication sequences are considered as particular cases. Moreover, some examples are given.     

On the minimum real roots of the adjoint polynomial of a graph

For a graph G, we denote by h(G,x) the adjoint polynomial of G. Let β(G) denote the minimum real root of h(G,x). In this paper, we characterize all the connected graphs G with .     

Euler-Mahonian statistics on ordered set partitions (II)

[E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the p,q-Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon's theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result.     

Tetracopper σ-Bound μ-Acetylide and -Diyne Units Stabilized by a Naphthyridine-based Dinucleating Ligand

Reactions of a dicopper(I) tert-butoxide complex with alkynes possessing boryl or silyl capping groups resulted in formation of unprecedented tetracopper(I) μ-acetylide/diyne complexes that were characterized by NMR and UV/Vis spectroscopy, mass spectrometry and single-crystal X-ray diffraction. These compounds possess an unusual μ₄-η¹:η¹:η¹:η¹ coordination mode for the bridging organic fragment, enforced by the rigid and dinucleating nature of the ligand utilized. Thus, the central π system remains unperturbed and accessible for subsequent reactivity and modification. This has been corroborated by addition of a fifth copper atom, giving rise to a pentacopper acetylide complex. This work may provide a new approach by which metal-metal cooperativity can be exploited in the transformation of acetylide and diyne groups to a variety of substrates, or as a starting point for the controlled synthesis of copper(I) alkyne-containing clusters.     

Dimerization of a Reactive Azaacene Diradical: Synthesis of a Covalent Azaacene Cage

Two series of regioisomeric dicyanomethylene substituted dithienodiazatetracenes with formal para- or ortho-quinodimethane subunits were synthesized and characterized. Whereas the para-isomers (p-n, diradical index y₀=0.01) are stable and isolable, the ortho-isomer (y₀=0.98) dimerizes into a covalent azaacene cage. Four elongated σ-CC bonds are formed and the former triisopropylsilyl(TIPS) -ethynylene groups transformed into cumulene units. The azaacene cage dimer (o-1)₂ was characterized by X-ray single crystal structure analysis and temperature-dependent infrared (IR), electron paramagnetic resonance (EPR, solid-state), nuclear magnetic resonance (NMR) and ultraviolet-visible (UV/Vis) spectroscopies (solution) indicating reformation of o-1.     

The Chemistry of the Cyaphide Ion

We review the known chemistry of the cyaphide ion, (C≡P)⁻. This remarkable diatomic anion has been the subject of study since the late nineteenth century, however its isolation and characterization eluded chemists for almost a hundred years. In this mini-review, we explore the pioneering synthetic experiments that first allowed for its isolation, as well as more recent developments demonstrating that cyaphide transfer is viable in well-established salt-metathesis protocols. The physical properties of the cyaphide ion are also explored in depth, allowing us to compare and contrast the chemistry of this ion with that of its lighter congener cyanide (an archetypal strong field ligand and important organic functional group). Recent studies show that the cyaphide ion has the potential to be used as a versatile chemical regent for the synthesis of novel molecules and materials, hinting at many interesting future avenues of investigation.     

Enhancing Carrier Transport via σ-Linkage Length Modulation in D-σ-A Semiconductors for Photocatalytic Oxidation

Carrier transport is an equally decisive factor as carrier separation for elevating photocatalytic efficiency. However, limited by indefinite structures and low crystallinities, studies on enhancing carrier transport of organic photocatalysts are still in their infancy. Here, we develop an σ-linkage length modulation strategy to enhance carrier transport in imidazole-alkyl-perylene diimide (IMZ-alkyl-PDI, corresponding to D-σ-A) photocatalysts by controlling π–π stacking distance. Ethyl-linkage can shorten π–π stacking distance (3.19 Å) the most among IMZ-alkyl-PDIs (where alkyl=none, ethyl, and n-propyl) via minimizing steric hindrance between D and A moieties, which leads to the fastest carrier transport rates. Thereby, IMZ-ethyl-PDI exhibits remarkable enhancement in phenol degradation with 32-fold higher rates than IMZ-PDI, as well as the oxygen evolution rate (271-fold increased). In microchannel reactors, IMZ-ethyl-PDI also presents 81.5 % phenol removal with high-flux surface hydraulic loading (44.73 L m⁻² h⁻¹). Our findings provide a promising molecular design guideline for high-performance photocatalysts and elucidate crucial internal carrier transport mechanisms.