%0 Journal Article
%T Welsh's problem on the number of bases of matroids
%A Sin Tsun Edward Fan
%A Wing Hong Tony Wong
%J Mathematics
%D 2014
%I arXiv
%X Given $(n,r,b)$, a triple of integers with $0<r\leq n$ and $1\leq b\leq\binom{n}{r}$, does there exist a matroid of size $n$, rank $r$, with $b$ bases? Such a matroid is called an $(n,r,b)$-matroid. This problem was raised by Dominic Welsh and is studied in this paper. We prove that when corank $n-r\leq3$, $(n,r,b)$-matroids exist for all such triples except $(n,r,b)=(6,3,11)$, which is the counterexample found by Mayhew and Royle. Furthermore, we show that $(n,r,b)$-matroids exist for all $r$ large relative to the corank $n-r$.
%U http://arxiv.org/abs/1402.6048v2