A Natural Family of Flag Matroids

Abstract

Abstract. A flag matroid can be viewed as a chain of matroids linked by quotients. Flag matroids, of which relatively few interesting families have previously been known, are a particular class of Coxeter matroids. In this paper we give a family of flag matroids arising from an enumeration problem that is a generalization of the tennis ball problem. These flag matroids can also be defined in terms of lattice paths and they provide a generalization of the lattice path matroids of [Bonin et al., JCTA 104 (2003)].

Lattice path matroids and quotients

“…Towards LPM flag diagrams. In [18] a certain class of (partial) LPFMs was studied, i.e., F : (M 0 , M 1 , . .…”

“…. , B k ) in F, one can associate a monotone path P of length n in Z k by setting the ith step to e This question is already present in [18,Figure 6], where an example shows that already pretty reasonable sets in Z 3 are not the diagram of an LPFM. We hope that the results of the present paper allow to shed new light on this problem.…”

“…This confirms a conjecture of Mcalmon, Oh, Xiang in the case of LPMs. We finish with diagram representations of LPFMs as suggested by de Mier [18], Higgs lifts and the weak order on LPMs, as well as questions on the structure of LPFMs within F ≥0 n .…”

“…The study of LPMs as a subclass of positroids, including analyzing quotients of these, is mostly novel apart from [18], where certain quotients of LPMs related to the tennis ball problem are explored.…”

Lattice path matroids: The excluded minors

“…A larger minor-closed class of transversal matroids was defined and studied in [3]. In [6], A. de Mier used lattice paths in higher dimensions to define a related type of flag matroid. Following a suggestion by V. Reiner [12] that there should be a type-B counterpart of the Catalan matroid (a certain nested matroid), J. Bonin and A. de Mier defined a class of Lagrangian matroids based on lattice paths; this topic has been studied by A. Gundert, E. Kim, and D. Schymura [8].…”