The multiple exchange property for matroid bases has recently been generalized for valuated matroids and M$^{ \natural}$-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of M$^{ \natural}$-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.
Categories: math.CO