Anti-lecture Hall Compositions and Overpartitions

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k-2 equals the number of overpartitions of n with non-overlined parts not congruent to $0,\pm 1$ modulo k. This identity can be considered as a refined version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartition which are analogous to the Rogers-Ramanjan type identities due to Andrews. When k is odd, we give an alternative proof by using a generalized Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and a refined version of a bijection also due to Corteel and Savage.Comment: 16 page

The Rogers-Ramanujan-Gordon Theorem for Overpartitions

Let $B_{k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{k,i}(n)=A_{k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruences condition. We show that $C_{k,i}(n)=D_{k,i}(n)$. By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of $D_{k,i}(n)$ equals the generating function of $C_{k,i}(n)$. We also find a generating function formula of $D_{k,i}(n)$ by using Gordon marking representations of overpartitions, which can be considered as an overpartition analogue of an identity of Andrews for ordinary partitions.Comment: 26 page

The JCMT BISTRO Survey: A Spiral Magnetic Field in a Hub-filament Structure, Monoceros R2

We present and analyze observations of polarized dust emission at 850 μm toward the central 1 7 1 pc hub-filament structure of Monoceros R2 (Mon R2). The data are obtained with SCUBA-2/POL-2 on the James Clerk Maxwell Telescope (JCMT) as part of the B-fields in Star-forming Region Observations survey. The orientations of the magnetic field follow the spiral structure of Mon R2, which are well described by an axisymmetric magnetic field model. We estimate the turbulent component of the magnetic field using the angle difference between our observations and the best-fit model of the underlying large-scale mean magnetic field. This estimate is used to calculate the magnetic field strength using the Davis–Chandrasekhar–Fermi method, for which we also obtain the distribution of volume density and velocity dispersion using a column density map derived from Herschel data and the C18O (J = 3 - 2) data taken with HARP on the JCMT, respectively. We make maps of magnetic field strengths and mass-to-flux ratios, finding that magnetic field strengths vary from 0.02 to 3.64 mG with a mean value of 1.0 \ub1 0.06 mG, and the mean critical mass-to-flux ratio is 0.47 \ub1 0.02. Additionally, the mean Alfv\ue9n Mach number is 0.35 \ub1 0.01. This suggests that, in Mon R2, the magnetic fields provide resistance against large-scale gravitational collapse, and the magnetic pressure exceeds the turbulent pressure. We also investigate the properties of each filament in Mon R2. Most of the filaments are aligned along the magnetic field direction and are magnetically subcritical

Magnetic Fields toward Ophiuchus-B Derived from SCUBA-2 Polarization Measurements

We present the results of dust emission polarization measurements of Ophiuchus-B (Oph-B) carried out using the Submillimetre Common-User Bolometer Array 2 (SCUBA-2) camera with its associated polarimeter (POL-2) on the James Clerk Maxwell Telescope in Hawaii. This work is part of the B-fields in Star-forming Region Observations survey initiated to understand the role of magnetic fields in star formation for nearby star-forming molecular clouds. We present a first look at the geometry and strength of magnetic fields in Oph-B. The field geometry is traced over ~0.2 pc, with clear detection of both of the sub-clumps of Oph-B. The field pattern appears significantly disordered in sub-clump Oph-B1. The field geometry in Oph-B2 is more ordered, with a tendency to be along the major axis of the clump, parallel to the filamentary structure within which it lies. The degree of polarization decreases systematically toward the dense core material in the two sub-clumps. The field lines in the lower density material along the periphery are smoothly joined to the large-scale magnetic fields probed by NIR polarization observations. We estimated a magnetic field strength of 630 ± 410 μG in the Oph-B2 sub-clump using a Davis–Chandrasekhar–Fermi analysis. With this magnetic field strength, we find a mass-to-flux ratio λ = 1.6 ± 1.1, which suggests that the Oph-B2 clump is slightly magnetically supercritical