159 research outputs found
The Catalan matroid
We show how the set of Dyck paths of length 2n naturally gives rise to a
matroid, which we call the "Catalan matroid" C_n. We describe this matroid in
detail; among several other results, we show that C_n is self-dual, it is
representable over the rationals but not over finite fields F_q with q < n-1,
and it has a nice Tutte polynomial.
We then generalize our construction to obtain a family of matroids, which we
call "shifted matroids". They arose independently and almost simultaneously in
the work of Klivans, who showed that they are precisely the matroids whose
independence complex is a shifted complex.Comment: 17 pages; submitted to the Journal of Combinatorial Theory - Series
The number of point-splitting circles
Let S be a set of 2n+1 points in the plane such that no three are collinear
and no four are concyclic. A circle will be called point-splitting if it has 3
points of S on its circumference, n-1 points in its interior and n-1 in its
exterior. We show the surprising property that S always has exactly n^2 point-
splitting circles, and prove a more general result.Comment: 12 pages, 4 figure
Flag arrangements and triangulations of products of simplices
We investigate the line arrangement that results from intersecting d complete
flags in C^n. We give a combinatorial description of the matroid T_{n,d} that
keeps track of the linear dependence relations among these lines. We prove that
the bases of the matroid T_{n,3} characterize the triangles with holes which
can be tiled with unit rhombi. More generally, we provide evidence for a
conjectural connection between the matroid T_{n,d}, the triangulations of the
product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d
tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and
effective criterion to ensure the vanishing of many Schubert structure
constants in the flag manifold, and a new perspective on Billey and Vakil's
method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo
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