Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the β invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the β invariant of certain lattice path matroids.

This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r : Z → Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.

We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds.The RGD algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert.The algorithm has been used to prove that there are no Eliahou semigroups of genus 66, hence proving the Wilf conjecture for genus up to 66. We also found three Eliahou semigroups of genus 67. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado's type. However, it is a member of a new family suggested by Shalom Eliahou.

Abstract. This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U -polynomial (or, equivalently, the same chromatic symmetric function). We consider the U kpolynomial, which is a restricted version of U -polynomial, and construct with the help of solutions of the Prouhet-Tarry-Escott problem, non-isomorphic trees with the same U k -polynomial for any given k. By doing so, we also find a new class of trees that are distinguished by the U -polynomial up to isomorphism.