Deutsch, Kitaev, and Remmel conjectured that two triples of permutation statistics are equidistributed over the symmetric group. One triple counts descents with odd tops, odd-odd adjacent pairs, and a statistic related to left-to-right maximal prefixes. The other triple rearranges these ingredients. The conjecture says the joint distribution over all permutations is identical for both triples.
Shankar resolves it bijectively. The proof constructs a recursive involution on permutations that swaps two of the statistics while preserving the third. The involution is built from a novel insertion process — a way of inserting elements into permutations that tracks the relevant statistics through the recursion.
The bijective approach matters because it does more than prove equality of distributions. It establishes a structural correspondence: each permutation with one statistic triple maps to a specific permutation with the other triple, and the map is its own inverse. The correspondence is not just existential but explicit — you can compute it.
The insertion process is the key technical contribution. Standard approaches to equidistribution conjectures often use generating function methods: compute both generating functions and show they're equal. This works for proving the identity but produces no bijection. Shankar's insertion process builds the bijection directly, using the recursive structure of permutations to define the involution one element at a time.
A conjecture from three combinatorialists. Resolved by one bijection. The map swaps what it must and fixes what it should.