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In a joint project with George Andrews, aspects of MacMahon's partition analysis have led us to consider broken partition diamonds, an infinite family of combinatorial objects whose generating functions give rise to a variety of number theoretic congruences. Recently, in the context of modular forms, Silviu Radu has set up an algorithmic machinery to prove such congruences automatically. The talk reports on recent developments, some being joint work with Radu, which combine methods from enumerative combinatorics and symbolic analysis (e.g. Riemann surfaces) with computer algebra.