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In 2020, Kang and Park conjectured a "level $$2$$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $$3$$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $$n$$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities. 

Abstract We demonstrate direct probing of strong magnon–photon coupling using Brillouin light scattering (BLS) spectroscopy in a planar geometry. The magnonic hybrid system comprises a split-ring resonator loaded with epitaxial yttrium iron garnet thin films of 200 nm and 2.46  μ m thickness. The BLS measurements are combined with microwave spectroscopy measurements where both biasing magnetic field and microwave excitation frequency are varied. The cooperativity for the 200 nm-thick YIG films is 1.1, and larger cooperativity of 29.1 is found for the 2.46 μ m-thick YIG film. We show that BLS is advantageous for probing the magnonic character of magnon–photon polaritons, while microwave absorption is more sensitive to the photonic character of the hybrid excitation. A miniaturized, planar device design is imperative for the potential integration of magnonic hybrid systems in future coherent information technologies, and our results are a first stepping stone in this regard. Furthermore, successfully detecting the magnonic hybrid excitation by BLS is an essential step for the up-conversion of quantum signals from the microwave to the optical regime in hybrid quantum systems. 

The direction of electron flow in molecular optoelectronic devices is dictated by charge transfer between a molecular excited state and an underlying conductor or semiconductor. For those devices, controlling the direction and reversibility of electron flow is a major challenge. We describe here a novel, single-molecule photodiode. It is based on an internally conjugated, bi-chromophoric dyad with chemically linked (porphyrinato)zinc(II) and bis(terpyridyl)ruthenium(II) groups. On nanocrystalline, degenerately doped indium tin oxide electrodes, the dyad exhibits distinct frequency-dependent, charge-transfer characters. Variations in the light source between red (~ 1.9 eV) and blue (~ 2.7 eV) light excitation for the integrated photodiode result in switching of photocurrents between cathodic and anodic. The origin of the excitation frequency-dependent photocurrents lies in the electronic structure of the chromophore excited states, as shown by the results of theoretical calculations, laser flash photolysis and steady-state spectrophotometric measurements.