Dual‐atom catalysts (DACs) have become an emerging platform to provide more flexible active sites for electrocatalytic reactions with multi‐electron/proton transfer, such as the CO2 reduction reaction (CRR). However, the introduction of asymmetric dual‐atom sites causes complexity in structure, leaving an incomprehensive understanding of the inter‐metal interaction and catalytic mechanism. Taking NiCu DACs as an example, herein, a more rational structural model is proposed, and the distance‐dependent inter‐metal interaction is investigated by combining theoretical simulations and experiments, including density functional theory computation, aberration‐corrected transmission electron microscopy, synchrotron‐based X‐ray absorption fine structure, and Monte Carlo experiments. A distance threshold around 5.3 Å between adjacent NiN4 and CuN4 moieties is revealed to trigger effective electronic regulation and boost CRR performance on both selectivity and activity. A universal macro‐descriptor rigorously correlating the inter‐metal distance and intrinsic material features (e.g., metal loading and thickness) is established to guide the rational design and synthesis of advanced DACs. This study highlights the significance of identifying the inter‐metal interaction in DACs, and helps bridge the gap between theoretical study and experimental synthesis of atomically dispersed catalysts with highly correlated active sites.

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Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions p −k (n) for all k ≥ 2. This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.

Abstract. Let p k (n) denote the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo 5 for p k (n). Moreover, we will present some new congruences modulo 7 for p4(n).

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