Enumeration of strong dichotomy patterns

We apply the version of P\'{o}lya-Redfield theory obtained by White to count patterns with a given automorphism group to the enumeration of strong dichotomy patterns, that is, we count bicolor patterns of $\mathbb{Z}_{2k}$ with respect to the action of \Aff(\mathbb{Z}_{2k}) and with trivial isotropy group. As a byproduct, a conjectural instance of phenomenon similar to cyclic sieving for special cases of these combinatorial objects is proposed.Comment: Some errors and unclear sentences had been correcte

Antichains and counterpoint dichotomies

We construct a special type of antichain (i. e., a family of subsets of a set, such that no subset is contained in another) using group-theoretical considerations, and obtain an upper bound on the cardinality of such an antichain. We apply the result to bound the number of strong counterpoint dichotomies up to affine isomorphisms

A Projection-Oriented Mathematical Model for Second-Species Counterpoint

Drawing inspiration from both the classical Guerino Mazzola's symmetry-based model for first-species counterpoint (one note against one note) and Johann Joseph Fux's "Gradus ad Parnassum", we propose an extension for second-species (two notes against one note)

The Diamine Cation Is Not a Chemical Example Where Density Functional Theory Fails

In a recent communication, Weber and co-workers presented a surprising study on charge-localization effects in the N,N'-dimethylpiperazine (DMP+) diamine cation to provide a stringent test of density functional theory (DFT) methods. Within their study, the authors examined various DFT methods and concluded that "all DFT functionals commonly used today, including hybrid functionals with exact exchange, fail to predict a stable charge-localized state." This surprising conclusion is based on the authors' use of a self-interaction correction (namely, complex-valued Perdew-Zunger Self-Interaction Correction (PZ-SIC)) to DFT, which appears to give excellent agreement with experiment and other wavefunction-based benchmarks. Since the publication of this recent communication, the same DMP+ molecule has been cited in numerous subsequent studies as a prototypical example of the importance of self-interaction corrections for accurately calculating other chemical systems. In this correspondence, we have carried out new high-level CCSD(T) analyses on the DMP+ cation to show that DFT actually performs quite well for this system (in contrast to their conclusion that all DFT functionals fail), whereas the PZ-SIC approach used by Weber et al. is the outlier that is inconsistent with the high-level CCSD(T) (coupled-cluster with single and double excitations and perturbative triples) calculations. Our new findings and analysis for this system are briefly discussed in this correspondence.Comment: Accepted by Nature Communication