1,110 research outputs found
History of Catalan numbers
We give a brief history of Catalan numbers, from their first discovery in the
18th century to modern times. This note will appear as an appendix in Richard
Stanley's forthcoming book on Catalan numbers.Comment: 10 page
The area of cyclic polygons: Recent progress on Robbins' Conjectures
In his works [R1,R2] David Robbins proposed several interrelated conjectures
on the area of the polygons inscribed in a circle as an algebraic function of
its sides. Most recently, these conjectures have been established in the course
of several independent investigations. In this note we give an informal outline
of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David
Robbins
A short proof of rigidity of convex polytopes
We present a much simplified proof of Dehn's theorem on the infinitesimal
rigidity of convex polytopes. Our approach is based on the ideas of Trushkina
and Schramm.Comment: to appear in Siberian Journal of Mathematics; 5 pages 2 figure
Non-commutative extensions of the MacMahon Master Theorem
We present several non-commutative extensions of the MacMahon Master Theorem,
further extending the results of Cartier-Foata and Garoufalidis-Le-Zeilberger.
The proofs are combinatorial and new even in the classical cases. We also give
applications to the -extension and Krattenthaler-Schlosser's
-analogue.Comment: 28 pages, 6 figure
On the complexity of computing Kronecker coefficients
We study the complexity of computing Kronecker coefficients
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of are also considered.Comment: v3: incorporated referee's comments; accepted to Computational
Complexit
Combinatorics and geometry of Littlewood-Richardson cones
We present several direct bijections between different combinatorial
interpretations of the Littlewood-Richardson coefficients. The bijections are
defined by explicit linear maps which have other applications.Comment: 15 pages, 9 figures. To be published in the special issue on
"Combinatorics and Representation Theory" of the European Journal of
Combinatoric
A Quantitative Steinitz Theorem for Plane Triangulations
We give a new proof of Steinitz's classical theorem in the case of plane
triangulations, which allows us to obtain a new general bound on the grid size
of the simplicial polytope realizing a given triangulation, subexponential in a
number of special cases.
Formally, we prove that every plane triangulation with vertices can
be embedded in in such a way that it is the vertical projection
of a convex polyhedral surface. We show that the vertices of this surface may
be placed in a integer grid, where and denotes the shedding diameter of , a
quantity defined in the paper.Comment: 25 pages, 6 postscript figure
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