134,989 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
Neighborliness of the symmetric moment curve
We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t,
sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges
over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long
as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the
convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize
the maximum possible length phi_k, proving, in particular, that phi_k > pi/2
for all k and that the limit of phi_k is pi/2 as k grows. This allows us to
construct centrally symmetric polytopes with a record number of faces.Comment: 28 pages, proofs are simplified and results are strengthened somewha
Expressing Forms as a sum of Pfaffians
Let A= (a_{ij}) be a symmetric non-negative integer 2k x 2k matrix. A is
homogeneous if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four
indexes. Let A be a homogeneous matrix and let F be a general form in C[x_1,
\dots x_n] with 2deg(F) = trace(A). We look for the least integer, s(A), so
that F= pfaff(M_1) + \cdots + pfaff(M_{s(A)}), where the M_i's are 2k x 2k
skew-symmetric matrices of forms with degree matrix A. We consider this problem
for n= 4 and we prove that s(A) < k+1 for all A
On the Multiple Packing Densities of Triangles
Given a convex disk and a positive integer , let and
denote the -fold translative packing density and the
-fold lattice packing density of , respectively. Let be a triangle.
In a very recent paper, K. Sriamorn proved that
. In this paper, I will show that
.Comment: arXiv admin note: text overlap with arXiv:1412.539
On Higher Dimensional Fuzzy Spherical Branes
Matrix descriptions of higher dimensional spherical branes are investigated.
It is known that a fuzzy 2k-sphere is described by the coset space
SO(2k+1)/U(k) and has some extra dimensions. It is shown that a fuzzy 2k-sphere
is comprised of n^{\frac{k(k-1)}{2}} spherical D(2k-1)-branes and has a fuzzy
2(k-1)-sphere at each point. We can understand the relationship between these
two viewpoints by the dielectric effect. Contraction of the algebra is also
discussed.Comment: 20 page
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