We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.2010 Mathematics Subject Classification. 05B35.

Recently Chase determined the maximum possible number of cliques of size t in a graph on n vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have m edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For s-graphs with s ≥ 3 a number of issues arise that do not appear in the graph case. For instance, for general s-graphs we can assign degrees to any i-subset of the vertex set with 1 ≤ i ≤ s − 1.We establish bounds on the number of t-cliques in an s-graph H with i-degree bounded by ∆ in three contexts: H has n vertices; H has m (hyper)edges; and (generalizing the previous case) H has a fixed number p of u-cliques for some u with s ≤ u ≤ t. When ∆ is of a special form we characterize the extremal s-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of Füredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.

Let ρ be a non-negative integer. A ρ-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of ρ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of ρ-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.