A greedoid and a matroid inspired by Bhargava's p-orderings
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Consider a finite set E equipped with a "weight function" w : E →R and a (symmetric) "distance function" d : E × E →R satisfying the ultrametric triangle inequality. We look for a subset of E of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances). We show that any such subset can be found by a greedy algorithm (which starts with the empty set, and then adds new elements one by one, maximizing the perimeter at each step). We use this to define numerical invariants, and also to show that the maximum-perimeter subsets of all sizes form a strong greedoid, and the maximum-perimeter subsets of any given size are the bases of a matroid. This essentially generalizes the "P-orderings" constructed by Bhargava in order to define his generalized factorials, and is also similar to the strong greedoid of maximum diversity subsets in phylogenetic trees studied by Moulton, Semple and Steel.
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