Mahonian Pairs
Bruce E. Sagan (Department of Mathematics, Michigan State University,, East Lansing, MI, USA), Carla D. Savage (Department of Computer Science,, North Carolina State University, Raleigh, NC, USA)

TL;DR
This paper introduces Mahonian pairs, explores their properties using Foata's bijection, and connects them with combinatorial structures like partitions, Catalan and Fibonacci numbers, and classical identities.
Contribution
It defines Mahonian pairs for finite and infinite sets, investigates their properties, and links them to well-known combinatorial identities and bijections.
Findings
Mahonian pairs generalize classical equidistribution results
Foata's bijection transforms word statistics into partition statistics
Connections established with Catalan, Fibonacci numbers, and Rogers-Ramanujan identities
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 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 different from T. Our principal tool is Foata's fundamental bijection f: P^* -> P^* since it has the property that maj w = inv f(w) for any word w. We consider various families of words associated with Catalan and Fibonacci numbers. We show that, when restricted to words in {1,2}^*, f transforms familiar statistics on words into natural statistics on integer partitions such as…
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Taxonomy
TopicsRomani and Gypsy Studies
