321-polygon-avoiding permutations and Chebyshev polynomials

A 321-k-gon-avoiding permutation pi avoids 321 and the following four patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1), k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k, (k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases k=2,3,4. In this paper, we extend these results by finding an explicit expression for the generating function for the number of 321-k-gon-avoiding permutations on n letters. The generating function is expressed via Chebyshev polynomials of the second kind.Comment: 11 pages, 1 figur

Repellence and attraction of Apis mellifera foragers by nectar alkaloids

Plant secondary metabolites present naturally in nectar, such as alkaloids, may change the behavioural responses of floral visitors and affect pollination. Some studies have shown that nectar containing low concentrations of these secondary metabolites is preferred by honey bee foragers over pure nectar. However, it remains unclear whether this is caused by dependence or addictive behaviour, a simple taste preference, or by other conditions such as self-medication. In our choice experiment, free-flying bees were presented with artificial flowers holding 20% sucrose containing 0.5−50 μg ml−1 of one of the naturally occurring nectar alkaloids - caffeine, nicotine, senecionine, and gelsemine. Nectar uptake was determined by weighing each flower and comparing the weight to that of the control flower. Our experimental design minimized memorizing and marking; despite this, caffeine was significantly preferred at concentrations 0.5−2 μg ml−1 over control nectar; this preference was not observed for other alkaloids. All of the compounds tested were repellent at concentrations above 5 μg ml−1. We confirmed previous reports that bees exhibit a preference for caffeine, and hypothesize that this is not due only to addictive behaviour but is at least partially mediated by taste preference. We observed no significant preference for nicotine or any other alkaloid

Finite Hilbert stability of (bi)canonical curves

We prove that a generic canonically or bicanonically embedded smooth curve has semistable m-th Hilbert points for all m. We also prove that a generic bicanonically embedded smooth curve has stable m-th Hilbert points for all m \geq 3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with G_m-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A_{2k+1}-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose m-th Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold. (This paper subsumes the previous submission and arXiv:1110.5960).Comment: To appear in Inventiones Mathematicae, 2012. The final publication is available at http://www.springerlink.co

Unshuffling Permutations: Trivial Bijections and Compositions

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