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

Herschel discovery of a new class of cold, faint debris discs

We present Herschel PACS 100 and 160 μm observations of the solar-type stars α Men, HD 88230 and HD 210277, which form part of the FGK stars sample of the Herschel open time key programme (OTKP) DUNES (DUst around NEarby Stars). Our observations show small infrared excesses at 160 μm for all three stars. HD 210277 also shows a small excess at 100 μm, while the 100 μm fluxes of α Men and HD 88230 agree with the stellar photospheric predictions. We attribute these infrared excesses to a new class of cold, faint debris discs. Both α Men and HD 88230 are spatially resolved in the PACS 160 μm images, while HD 210277 is point-like at that wavelength. The projected linear sizes of the extended emission lie in the range from ~115 to ≤ 250 AU. The estimated black body temperatures from the 100 and 160 μm fluxes are ≲22 K, and the fractional luminosity of the cold dust is L_(dust)/L_⋆ ~ 10^(-6), close to the luminosity of the solar-system’s Kuiper belt. These debris discs are the coldest and faintest discs discovered so far around mature stars, so they cannot be explained easily invoking “classical” debris disc models