Séminaire Lotharingien de Combinatoire, 95B.70 (2026), 12 pp.
Arvind Ayyer and Naren Sundaravaradan
A Bijection on Balanced Words Reversing both des and maj
Abstract.
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.
We give an explicit bijection proving this result.
We also show that for balanced binary words, this bijection simultaneously flips the ascent and comajor index as well.
For permutations (which are also examples of balanced words), we show that our bijection is different from the complementation map.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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