46 research outputs found
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 0.5â50 ÎŒg mlâ1 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 0.5â2 ÎŒg mlâ1 over control nectar; this preference was not observed for other alkaloids. All of the compounds tested were repellent at concentrations above 5 ÎŒg mlâ1. 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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