148,141 research outputs found
A bijection to count (1-23-4)-avoiding permutations
A permutation is (1-23-4)-avoiding if it contains no four entries, increasing
left to right, with the middle two adjacent in the permutation. Here we give a
2-variable recurrence for the number of such permutations, improving on the
previously known 4-variable recurrence. At the heart of the proof is a
bijection from (1-23-4)-avoiding permutations to increasing ordered trees whose
leaves, taken in preorder, are also increasing.Comment: latex, 16 page
A new approach to nonrepetitive sequences
A sequence is nonrepetitive if it does not contain two adjacent identical
blocks. The remarkable construction of Thue asserts that 3 symbols are enough
to build an arbitrarily long nonrepetitive sequence. It is still not settled
whether the following extension holds: for every sequence of 3-element sets
there exists a nonrepetitive sequence with
. Applying the probabilistic method one can prove that this is true
for sufficiently large sets . We present an elementary proof that sets of
size 4 suffice (confirming the best known bound). The argument is a simple
counting with Catalan numbers involved. Our approach is inspired by a new
algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its
interpretations by Fortnow and Tao. The presented method has further
applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with
arXiv:1103.381
The algebra of cell-zeta values
In this paper, we introduce cell-forms on , which are
top-dimensional differential forms diverging along the boundary of exactly one
cell (connected component) of the real moduli space
. We show that the cell-forms generate the
top-dimensional cohomology group of , so that there is a
natural duality between cells and cell-forms. In the heart of the paper, we
determine an explicit basis for the subspace of differential forms which
converge along a given cell . The elements of this basis are called
insertion forms, their integrals over are real numbers, called cell-zeta
values, which generate a -algebra called the cell-zeta algebra. By
a result of F. Brown, the cell-zeta algebra is equal to the algebra of
multizeta values. The cell-zeta values satisfy a family of simple quadratic
relations coming from the geometry of moduli spaces, which leads to a natural
definition of a formal version of the cell-zeta algebra, conjecturally
isomorphic to the formal multizeta algebra defined by the much-studied double
shuffle relations
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur
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