The structure of the consecutive pattern poset

The consecutive pattern poset is the infinite partially ordered set of all permutations where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent entries in the same relative order as the entries of $\sigma$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR

Increasing spanning forests in graphs and simplicial complexes

Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F forms an increasing sequence. Hallam and Sagan showed that the generating function ISF(G, t) for increasing spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,..., n is a perfect elimination order for G. We give new, purely combinatorial proofs of these results which permit us to generalize them in several ways. For example, we are able to bound the coef- cients of ISF(G, t) using broken circuits. We are also able to extend these results to simplicial complexes using the new notion of a cage-free complex. A generalization to labeled multigraphs is also given. We observe that the de nition of an increasing spanning forest can be formulated in terms of pattern avoidance, and we end by exploring spanning forests that avoid the patterns 231, 312 and 321

Seiberg-Witten Floer K-theory and cyclic group actions on spin four-manifolds with boundary

Given a spin rational homology sphere $Y$ equipped with a $\mathbb{Z}/m$-action preserving the spin structure, we use the Seiberg--Witten equations to define equivariant refinements of the invariant $\kappa(Y)$ from \cite{Man14}, which take the form of a finite subset of elements in a lattice constructed from the representation ring of a twisted product of $\text{Pin}(2)$ and $\mathbb{Z}/m$. The main theorems consist of equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres. We provide applications to knot concordance, give obstructions to extending cyclic group actions to spin fillings, and via taking branched covers we obtain genus bounds for knots in punctured 4-manifolds. In some cases, these bounds are strong enough to determine the relative genus for a large class of knots within certain homology classes in $\mathbb{C} P^{2}\#\mathbb{C} P^{2}$, $S^{2}\times S^{2}\# S^{2}\times S^{2}$, $\mathbb{C} P^{2}\# S^{2}\times S^{2}$, and homotopy $K3$ surfaces.Comment: v2: Edits for clarity, changes in notation, etc. Significant rewrites of sections 8.2 and 8.3; fixed calculations for higher order action