  We exhibit a bijection between recently-introduced combinatorial objects
known as valid hook configurations and certain weighted set partitions. When
restricting our attention to set partitions that are matchings, we obtain three
new combinatorial interpretations of Lassalle's sequence. One of these
interpretations involves permutations that have exactly one preimage under the
(West) stack-sorting map. We prove that the sequences obtained by counting
these permutations according to their first entries are symmetric, and we
conjecture that they are log-concave. We also obtain new recurrence relations
involving Lassalle's sequence and the sequence that enumerates valid hook
configurations. We end with several suggestions for future work.
