Balanced words on a finite alphabet are those words in which every letter of the alphabet occurs the same number of times. The notion of descents and major index extends in a natural way to words. It is known that the bivariate generating polynomials for descents and major index over balanced words on the alphabet $[k]$ with $n$ occurrences each is palindromic, but a bijective proof has been missing even for balanced binary words. In this talk, I will first present a bijection for balanced binary words and then recursively extend the bijection for all balanced words. I will also describe some of the properties of this bijection on restrictions of balanced words like permutations.

This is a joint work with Arvind Ayyer (IISc).