810 research outputs found
Symmetric unimodal expansions of excedances in colored permutations
We consider several generalizations of the classical -positivity of
Eulerian polynomials (and their derangement analogues) using generating
functions and combinatorial theory of continued fractions. For the symmetric
group, we prove an expansion formula for inversions and excedances as well as a
similar expansion for derangements. We also prove the -positivity for
Eulerian polynomials for derangements of type . More general expansion
formulae are also given for Eulerian polynomials for -colored derangements.
Our results answer and generalize several recent open problems in the
literature.Comment: 27 pages, 10 figure
On two unimodal descent polynomials
The descent polynomials of separable permutations and derangements are both
demonstrated to be unimodal. Moreover, we prove that the -coefficients
of the first are positive with an interpretation parallel to the classical
Eulerian polynomial, while the second is spiral, a property stronger than
unimodality. Furthermore, we conjecture that they are both real-rooted.Comment: 16 pages, 4 figure
Hair-brane Ideas on the Horizon
We continue an examination of the microstate geometries program begun in
arXiv:1409.6017, focussing on the role of branes that wrap the cycles which
degenerate when a throat in the geometry deepens and a horizon forms. An
associated quiver quantum mechanical model of minimally wrapped branes exhibits
a non-negligible fraction of the gravitational entropy, which scales correctly
as a function of the charges. The results suggest a picture of AdS_3/CFT_2
duality wherein the long string that accounts for BTZ black hole entropy in the
CFT description, can also be seen to inhabit the horizon of BPS black holes on
the gravity side.Comment: 50 pages, 4 figures. v2: minor corrections, reference adde
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