T. Kyle Petersen

The γ-vector of a barycentric subdivision

We prove that the @c-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.

On γ -Vectors Satisfying the Kruskal–Katona Inequalities

We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative.

Two-Sided Eulerian Numbers via Balls in Boxes

Summary The Eulerian numbers count permutations according to the number of descents. The two-sided Eulerian numbers count permutations according to number of descents and the number of descents in the inverse permutation. Here we derive some results for Eulerian and two-sided Eulerian numbers using an elementary “balls-in-boxes” approach. We also discuss an open conjecture of Ira Gessel about the two-sided Eulerian numbers.

On the Shard Intersection Order of a Coxeter Group

Introduced by Reading, the shard intersection order of a finite Coxeter group

Computing reflection length in an affine Coxeter group

W

Characterizing f-vectors (Supplemental)

is a lattice structure on the elements of

Weak order, hyperplane arrangements, and the Tamari lattice

W

W-Narayana numbers

that contains the poset of noncrossing partitions

Cubes, Carries, and an Amazing Matrix (Supplemental)

NC(W)

Affine descents and the Steinberg torus (Supplemental)

as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models for shard intersections of all classical types and use this understanding to prove that the shard intersection order is EL-shellable. Further, inspired by work of Simion and Ullman on the lattice of noncrossing partitions, we show that the shard intersection order on the symmetric group admits a symmetric Boolean decomposition, i.e., a partition into disjoint Boolean algebras whose middle ranks coincide with the middle rank of the poset. Our decomposition also yields a new symmetric Boolean decomposition of the noncrossing partition lattice.