Large families of measures on partitions share the same moments; the p=2 sandpile distributions of Mészáros and of random bipartite graphs both sit inside one such family.
On a Cohen–Lenstra heuristic for Jacobians of random graphs.Journal of Algebraic Combinatorics, 42(3):701–723, 2015
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Sandpile groups of random bipartite graphs and families of distributions with the same moments
Large families of measures on partitions share the same moments; the p=2 sandpile distributions of Mészáros and of random bipartite graphs both sit inside one such family.