Permutation tableaux and permutation patterns

In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the -diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz' q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot.     

Synthesis and structural characterization of cross-linked histidine-phenol Cu(ii) complexes as cytochrome c oxidase active site models

Tridentate cross-linked histidine-phenol Cu(ii) ether and ester complexes, chemical analogs of the active site of several heme-copper oxidases, have been synthesized and crystallized.     

Reactions of CF3CH2I+O(3P): competing mechanisms of HF elimination

Ab initio density functional and molecular orbital calculations provide singlet and triplet electronic potential energy surfaces for the reactions of CF3CH2I+O(3P) leading to OI and HF eliminations, reactions which have been the subject of recent experimental studies. A barrier to OI formation occurs on the triplet potential energy surface; there is no reverse barrier to OI formation on the singlet pathway. Findings suggest that two competing pathways may form HF. One is an addition-insertion-elimination process involving insertion of O into the C-I bond. The alternate path involves OI elimination, addition of an O atom to CF3CH2, and subsequent HF elimination. The computed reactant pathways and energetics are discussed in relation to recent experiments.     

Copper(I)-mediated and microwave-assisted C(aryl)-O(carboxylic) coupling: synthesis of benzopyranones and isolamellarin alkaloids

A simple and highly effective C-O(carboxylic) coupling reaction catalyzed by copper(I) salts has been developed to synthesize benzopyranones. The reaction of various 2-halobiarylcarboxylic acids was examined using microwave irradiation. A new class of pyrroloisoquinoline alkaloid, isolamellarin, was synthesized based on the annulation of dihydroisoquinoline with aryl pyruvates under basic condition and Cu-mediated/MW-assisted C-O(carboxylic) lactonization.     

Welwistatin support studies: expansion and limitation of aryllead(IV) coupling reactions

Recent support studies on the total synthesis of the welwistatin system are described. The target step involves lead-mediated arylation of sterically demanding aryl groups and carbon acid coupling partners in order to establish the highly congested tetracyclic core structure. Type 7 beta-ketoesters and beta-ketonitriles were successfully arylated with a variety of ortho- and meta-substituted aryllead compounds generated by a halogen-boron-lead exchange sequence. The enolates of compounds 15, 19, and 25, each bearing all-carbon quaternary centers adjacent to the arylation site, failed to couple.     

Poly­[[aqua­neodymium(III)]‐μ3‐decane‐1,10‐di­carboxyl­ato‐μ3‐9‐carboxy­nonane­carboxyl­ato]

The title compound, [Nd(C₁₀H₁₆O₄)(C₁₀H₁₇O₄)(H₂O)]_n, has a novel Nd–organic framework constructed from sebacic acid (C₁₀H₁₈O₄) linkers, the longest aliphatic ligand used to date in lanthanide metal–organic framework compounds. The structure contains edge‐shared chains of NdO₈(H₂O) tricapped trigonal prisms that propagate in the [100] direction, with Nd—O distances in the range 2.414 (4)–2.643 (4) Å.     

Synthesis of 2,6-disubstituted dihydropyrans via an efficient BiBr3-initiated three component, one-pot cascade

The rapid synthesis of cis-2,6-disubstituted dihydropyrans is achieved in a three-component, one-pot cascade reaction. BiBr₃-mediated addition of ketene silyl acetals or silyl enol ethers to β,γ-unsaturated cis-4-trimethylsilyl-3-butenal provides a Mukaiyama aldol adduct containing a vinylsilane moiety tethered to a silyl ether. Addition of a second aldehyde initiates a domino sequence involving intermolecular addition followed by an intramolecular silyl-modified Sakurai (ISMS) reaction. Isolated yields of this one-pot reaction vary from 44 to 80% and all compounds were isolated as the cis-diastereomers (10 examples).     

Development of a Fluorescent Pb2+ Sensor

A real turn‐on : The emission intensity of heterocycle 1 increases upon binding to Pb²⁺. Thus, 1 acts as a small‐molecule “turn‐on” fluorescent sensor for lead. The sensor is highly selective and is functional over a wide range of pH values.

     

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.