Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$

Highly selective and solvent-dependent reduction of Nitrobenzene to N-phenylhydroxylamine, azoxybenzene, and aniline catalyzed by phosphino-modified polymer immobilized ionic liquid-stabilized AuNPs

Gold nanoparticles stabilized by phosphine-decorated polymer immobilized ionic liquids (AuNP@PPh2-PIILP) is an extremely efficient multiproduct selective catalyst for the sodium borohydride-mediated reduction of nitrobenzene giving N-phenylhydroxylamine, azoxybenzene, or aniline as the sole product under mild conditions and a very low catalyst loading. The use of a single nanoparticle-based catalyst for the partial and complete reduction of nitroarenes to afford three different products with exceptionally high selectivities is unprecedented. Under optimum conditions, thermodynamically unfavorable N-phenylhydroxylamine can be obtained as the sole product in near quantitative yield in water, whereas a change in reaction solvent to ethanol results in a dramatic switch in selectivity to afford azoxybenzene. The key to obtaining such a high selectivity for N-phenylhydroxylamine is the use of a nitrogen atmosphere at room temperature as reactions conducted under an inert atmosphere occur via the direct pathway and are essentially irreversible, while reactions in air afford significant amounts of azoxy-based products by virtue of competing condensation due to reversible formation of N-phenylhydroxylamine. Ultimately, aniline can also be obtained quantitatively and selectively by adjusting the reaction temperature and time accordingly. Introduction of PEG onto the polyionic liquid resulted in a dramatic improvement in catalyst efficiency such that N-phenylhydroxylamine could be obtained with a turnover number (TON) of 100 000 (turnover frequency (TOF) of 73 000 h–1, with >99% selectivity), azoxybenzene with a TON of 55 000 (TOF of 37 000 h–1 with 100% selectivity), and aniline with a TON of 500 000 (TOF of 62 500 h–1, with 100% selectivity). As the combination of ionic liquid and phosphine is required to achieve high activity and selectivity, further studies are currently underway to explore whether interfacial electronic effects influence adsorption and thereby selectivity and whether channeling of the substrate by the electrostatic potential around the AuNPs is responsible for the high activity. This is the first report of a AuNP-based system that can selectively reduce nitroarenes to either of two synthetically important intermediates as well as aniline and, in this regard, is an exciting discovery that will form the basis to develop a continuous flow process enabling facile scale-up