The Eulerian Distribution on Involutions is Indeed Unimodal

Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$\sum_{k=0}^{n-1}I_{n,k}t^k=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}.$$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.Comment: 12 pages, minor changes, to appear in J. Combin. Theory Ser.

The Early Evolution of Massive Stars: Radio Recombination Line Spectra

Velocity shifts and differential broadening of radio recombination lines are used to estimate the densities and velocities of the ionized gas in several hypercompact and ultracompact HII regions. These small HII regions are thought to be at their earliest evolutionary phase and associated with the youngest massive stars. The observations suggest that these HII regions are characterized by high densities, supersonic flows and steep density gradients, consistent with accretion and outflows that would be associated with the formation of massive stars.Comment: ApJ in pres

Resolving the Structure and Kinematics of the Youngest HII Regions and Radio Jets from Young Stellar Objects

In this contribution we explore the new science that a Next Generation Very Large Array (ngVLA) would be able to perform on the topics of the youngest HII regions and (proto)stellar jets. Free-free continuum and radio recombination line (RRL) emission are often the only way of peering into the dense envelopes surrounding (proto)stars of all masses, and trace their initial feedback in the form of `radio jets', `hypercompact HII regions', or photoevaporating, partially-ionized flows. Properly disentangling free-free from dust emission is also mandatory in studies of protoplanetary and accretion disks. Current VLA research has reached an impasse in which a population of faint ionized radio sources, probably corresponding to the above mentioned objects, is detected, but their nature is mostly unknown. The ngVLA would allow us to resolve the density structure and kinematics of such sources, revolutionizing our knowledge of star formation across the entire stellar-mass spectrum.Comment: An ngVLA Science Book chapter. v2 includes a few modifications in reference