For each composition c we show that the order complex of the poset of pointed set partitions Π • c is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module S B where B is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study facial structures of matroid base polytopes corresponding to lattice path matroids.

We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we show how to find the equation determining the spectrum. We give two length $4$ applications. First, we find the asymptotics of the number of permutations with no triple ascents and no triple descents. Second, we give the asymptotics of the number of permutations with no isolated ascents or descents. Our next result is a weighted pattern of length $3$ where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.Comment: 16 page

Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are n items and m servers, each of which stores a subset of the items. A batch code is an arrangement for storing items on servers so that, for prescribed integers k and t, any k items can be retrieved by reading at most t items from each server. Silberstein defined an erasure batch code (with redundancy r) as a batch code in which any k items can be retrieved by reading at most t items from each server, while any r servers are unavailable (failed).In this paper, we investigate erasure batch codes with t = 1 (each server can read at most one item) in a combinatorial manner. We determine the optimal (minimum) total storage of an erasure batch code for several ranges of parameters. Additionally, we relate optimal erasure batch codes to maximum packings. We also identify a necessary lower bound for the total storage of an erasure batch code, and we relate parameters for which this trivial lower bound is achieved to the existence of graphs with appropriate girth.

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