Mahonian Pairs

We introduce the notion of a Mahonian pair. Consider the set, P*, of all words having the positive integers as alphabet. Given finite subsets S,T of P*, we say that (S,T) is a Mahonian pair if the distribution of the major index, maj, over S is the same as the distribution of the inversion number, inv, over T. So the well-known fact that maj and inv are equidistributed over the symmetric group, S_n, can be expressed by saying that (S_n,S_n) is a Mahonian pair. We investigate various Mahonian pairs (S,T) with S≠T. Our principal tool is Foata's fundamental bijection φ:P*→P* since it has the property that maj(w) = inv(φ(w)) for any word w. We consider various families of words associate with Catalan and Fibonacci numbers. A number of ideas come into play such as the ranks and Durfee square size of integer partitions, the Catalan triangle, and certrain q-analogues.

This is joint work with Carla Savage.