Quadrant marked mesh patterns in 123-avoiding permutations

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions

Static and Dynamical Properties of the Spin-1/2 Equilateral Triangular-Lattice Antiferromagnet Ba3_3CoSb2_2O9_9

We present single-crystal neutron scattering measurements of the spin-1/2 equilateral triangular lattice antiferromagnet Ba$_3$CoSb$_2$O$_9$. Besides confirming that the Co$^{2+}$ magnetic moments lie in the ab plane for zero magnetic field, we determine all the exchange parameters of the minimal quasi-2D spin Hamiltonian, which confirms that Ba$_3$CoSb$_2$O$_9$ is an almost perfect realization of the paradigmatic model of frustrated quantum magnetism. A comparison with linear and nonlinear spin-wave theory reveals that quantum fluctuations induce a strong downward renormalization of the magnon dispersion.Comment: 8 pages, 7 figures, with supplementar