The University of Michigan Combinatorics Seminar


Abstract 

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 wellknown
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 qanalogues.
