If one insists, the transposition map on the set of partitions of a given integer may be viewed as a Langlands reciprocity phenomenon. By altering the symmetry type, this leads to the Barbasch–Vogan–Lusztig–Spaltenstein duality, which relates orthogonal and symplectic partitions. While this duality holds in itself the intricate geometry of non-simply-connected nilpotent orbits in classical Lie algebras, it remains combinatorially accessible. The goal of the talk will be to offer a taste of this theme without assuming too much background, while surveying a recent result with Emile Okada regarding decompositions of Springer representations.