Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.
Strongly correlated catalysts can be understood from precise quantum approximations. Incorporating properly electronic correlations thus let’s define Spin rules in catalysis, opening a new door towards optimum compositions for the most important reactions for a sustainable future. 相似文献
Journal of Solid State Electrochemistry - The potential dependence of the wetting angle on a gold electrode in sodium, potassium, and cesium bromide and iodide melts was studied to establish the... 相似文献
The carbon cage of buckminsterfullerene Ih-C60, which obeys the Isolated-Pentagon Rule (IPR), can be transformed to non-IPR cages in the course of high-temperature chlorination of C60 or C60Cl30 with SbCl5. The non-IPR chloro derivatives were isolated chromatographically (HPLC) and characterized crystallographically as 1809C60Cl16, 1810C60Cl24, and 1805C60Cl24, which contain, respectively two, four, and four pairs of fused pentagons in the carbon cage. High-temperature trifluoromethylation of the chlorination products with CF3I afforded a non-IPR CF3 derivative, 1807C60(CF3)12, which contains four pairs of fused pentagons in the carbon cage. Addition patterns of non-IPR chloro and CF3 derivatives were compared and discussed in terms of the formation of stabilizing local substructures on fullerene cages. A detailed scheme of the experimentally confirmed non-IPR C60 isomers obtained by Stone–Wales cage transformations is presented. 相似文献