Sizes of flats of cycle matroids of complete graphs

Abstract

We show that the problem of counting the number of flats of size k for a cycle matroid of a complete graph is equivalent to the problem of counting the number of partitions of an integer k into triangular numbers. In addition, we give some values of k such that there is no flat of size k in a cycle matroid of a complete graph of order k. Finally, we give a minimum bound for the number of values, k, for which there are no flats of size k in the given cycle matroid.