The q-factorial in terms of the major index

EDIT: As pointed out in the comments, the following proof is found in Knuth’s book, The Art of Computer Programming, Vol. 3, and is attributed to MacMahon.

There are several nice series expansions involving basic permutation statistics which I’ve been reviewing lately. One of these involves the number of inversions of a permutation.

An inversion of a permutation of the numbers $1,2,\ldots,n$ is a pair of numbers which appear “out of order” in the sequence, that is, with the larger number appearing first. For instance, for $\DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\inv}{inv} n=4$, the permutation $3,1,4,2$ has three inversions: $(3,1)$, $(3,2)$, and $(4,2)$. We write $$\inv(3142)=3.$$

It is not hard to show that
$$(1)(1+q)(1+q+q^2)\cdots(1+q+q^2+\cdots+q^{n-1})=\sum q^{\inv(w)}$$ where the sum ranges over all permutations $w$ of $\{1,2,\ldots,n\}$. (See page 3 of this post for a proof.) The product on the left side is called the $n$th $q$-factorial.

The remarkable thing is that there is a similar expansion of the $q$-factorial in terms of the major index. Define a descent of a permutation to be an index $i$ for which the $i$th number is greater than the $(i+1)$st. The major index is defined to be the sum of the descents. For instance, the permutation $3,1,4,2$ has two descents, in positions $1$ and $3$, so the major index is $4$. We write $$\maj(3142)=4.$$

With this notation, we have
\begin{equation}(1)(1+q)(1+q+q^2)\cdots(1+q+q^2+\cdots+q^{n-1})=\sum_{w\in S_n} q^{\maj(w)}\end{equation}
as well, implying that the number of permutations with $k$ inversions is equal to the number of permutations with major index $k$, for any $k$. There is a direct bijection showing this called the Foata bijection.

I was seeking a way to prove equation (1) directly, without the Foata bijection. But the simplest proof I was able to find in the literature, in Stanley’s Enumerative Combinatorics, proved it by showing something much more general and then simply stating it as a corollary. Wishing to see a direct proof of the result, I went through Stanley’s more general proof step-by step to see what was going on in this special case. I thought I would share it here since I had a hard time finding it elsewhere. (Turn to page 2.)

2 thoughts on “The q-factorial in terms of the major index

  1. This proof can also be found in Knuth’s The Art of Computer Programming Vol.3 5.1.1, where it is attributed to MacMahon.

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