Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A and s_B are equal, then the skew shapes A and B must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than skew Schur equality: that s_A and s_B have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of s_A contains that of s_B, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.Comment: 26 pages, 7 figures. J. Combin., to appear. Version 2 includes a new subsection (5.3) on a possible skew version of the Saturation Theore

On the topology of the permutation pattern poset

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the M\"obius function of decomposable permutations given by Burstein et al.Comment: 33 pages, 4 figures. Incorporates changes suggested by the referees; new open problems in Subsection 9.4. To appear in JCT(A

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

The M\"obius function of generalized subword order

Let P be a poset and let P* be the set of all finite length words over P. Generalized subword order is the partial order on P* obtained by letting u \leq w if and only if there is a subword u' of w having the same length as u such that each element of u is less than or equal to the corresponding element of u' in the partial order on P. Classical subword order arises when P is an antichain, while letting P be a chain gives an order on compositions. For any finite poset P, we give a simple formula for the Mobius function of P* in terms of the Mobius function of P. This permits us to rederive in a easy and uniform manner previous results of Bjorner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in P* for any finite P of rank at most 1.Comment: 29 pages, 4 figures. Incorporates referees' suggestions; to appear in Advances in Mathematic

Equality of P-partition generating functions

To every labeled poset (P,\omega), one can associate a quasisymmetric generating function for its (P,\omega)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,\omega) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of linear extensions.Comment: 24 pages, 19 figures. Incorporates minor changes suggested by the referees. To appear in Annals of Combinatoric

Active galactic nucleus feedback in clusters of galaxies

Observations made during the last ten years with the Chandra X-ray Observatory have shed much light on the cooling gas in the centers of clusters of galaxies and the role of active galactic nucleus (AGN) heating. Cooling of the hot intracluster medium in cluster centers can feed the supermassive black holes found in the nuclei of the dominant cluster galaxies leading to AGN outbursts which can reheat the gas, suppressing cooling and large amounts of star formation. AGN heating can come in the form of shocks, buoyantly rising bubbles that have been inflated by radio lobes, and the dissipation of sound waves.Comment: Refereed review article published in Chandra's First Decade of Discovery Special Feature edition of the Proceedings of the National Academy of Science