[Ni(NHC)2] as a Scaffold for Structurally Characterized trans [H−Ni−PR2] and trans [R2P−Ni−PR2] Complexes

The addition of PPh₂H, PPhMeH, PPhH₂, P(para-Tol)H₂, PMesH₂ and PH₃ to the two-coordinate Ni⁰ N-heterocyclic carbene species [Ni(NHC)₂] (NHC=IiPr₂, IMe₄, IEt₂Me₂) affords a series of mononuclear, terminal phosphido nickel complexes. Structural characterisation of nine of these compounds shows that they have unusual trans [H−Ni−PR₂] or novel trans [R₂P−Ni−PR₂] geometries. The bis-phosphido complexes are more accessible when smaller NHCs (IMe₄>IEt₂Me₂>IiPr₂) and phosphines are employed. P−P activation of the diphosphines R₂P−PR₂ (R₂=Ph₂, PhMe) provides an alternative route to some of the [Ni(NHC)₂(PR₂)₂] complexes. DFT calculations capture these trends with P−H bond activation proceeding from unconventional phosphine adducts in which the H substituent bridges the Ni−P bond. P−P bond activation from [Ni(NHC)₂(Ph₂P−PPh₂)] adducts proceeds with computed barriers below 10 kcal mol⁻¹. The ability of the [Ni(NHC)₂] moiety to afford isolable terminal phosphido products reflects the stability of the Ni−NHC bond that prevents ligand dissociation and onward reaction.     

Synthesis of lodopyridone

The total synthesis of the unusual 4-pyridone marine metabolite lodopyridone has been achieved by late stage manipulation of the related 4-pyrone. Key reactions include a Suzuki–Miyaura coupling to form the tetracyclic core and a modified Corey–Ganem–Gilman reaction to install the ethanolamide side-chain.     

A major index for matchings and set partitions

We introduce a statistic pmaj(P) for partitions of [n], and show that it is equidistributed with cr₂, the number of 2-crossings, over all partitions of [n] with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inv and maj.     

Linked partitions and linked cycles

The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the n-th large Schröder number r_n, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, k-Stirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs.     

Stahl

Ohne Zusammenfassung     

Ueber die quantitative Bestimmung und Trennung der Cacao-Alkaloide

Ohne Zusammenfassung     

Some identities between basic hypergeometric series deriving from a new Bailey-type transformation

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.     

On differential Rota-Baxter algebras

A Rota-Baxter operator of weight λ is an abstraction of both the integral operator (when λ=0) and the summation operator (when λ=1). We similarly define a differential operator of weight λ that includes both the differential operator (when λ=0) and the difference operator (when λ=1). We further consider an algebraic structure with both a differential operator of weight λ and a Rota-Baxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.