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 \[\inv(3142)=3.\]

Growth patterns of desert bushes?

I spent most of Spring Break hiking in Death Valley National Park with Bryan Gillespie. It was stunningly beautiful and rather otherworldly in some ways, especially in the way that so much hardy desert life keeps on surviving on 1.5 inches of rainfall per year.

One such hardy desert plant that we encountered a number of times, especially near the Ubehebe Crater, was the creosote bush. It looked like this:

Bryan’s first reaction was, “Wow, look at the way it grows! I’d love to know what its fractal structure is.” He walked around the plant and took several more photos:

It appeared that whatever angle you viewed the bush from, there was (approximately) a single layer of leaves visible with no leaf covering another and no big gaps between the leaves. This, we realized, is an optimal use of its resources: no matter the angle of the sun, it is using every leaf to absorb its energies while not wasting water keeping other buried leaves alive.

A q-analog of the decomposition of the regular representation of the symmetric group

It is a well-known fact of representation theory that, if the irreducible representations of a finite group $\DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\sh}{sh} G$ are $V_1,\ldots,V_m$, and $R$ is the regular representation formed by $G$ acting on itself by left multiplication, then \[R=\bigoplus_{i=1}^{m} (\dim V_i) \cdot V_i\] is its decomposition into irreducibles.

I’ve recently discovered a $q$-analog of this fact for $G=S_n$ that is a simple consequence of some known results in symmetric function theory.

In Enumerative Combinatorics, Stanley defines a generalization of the major index on permutations to standard tableaux. For a permutation \[w=w_1,\ldots,w_n\] of $1,\ldots,n$, a descent is a position $i$ such that $w_i>w_{i+1}$. For instance, $52413$ has two descents, in positions $1$ and $3$. The major index of $w$, denoted $\maj(w)$, is the sum of the positions of the descents, in this case \[\maj(52413)=1+3=4.\]

The Mad Veterinarian

I volunteered at the Julia Robinson Math Festival held at a nearby school last Saturday. It was a great event for kids in 4th to 9th grade in the area, featuring a room full of mathematical activity tables and raffle tickets to be given out as rewards for a good idea or explanation.

The table I was working at featured a number of problems about a ``mad veterinarian’’. He had a machine that you can feed two cats to and get a dog out. You can also feed it two dogs to get one dog out, and if you feed it a cat and a dog, you get out a cat. The veterinarian wants to end up with only one cat, and no other animals. Can he do this starting with three cats and one dog? How about four cats and two dogs?

Rota's Indiscrete Thoughts

I am a huge fan of Gian-Carlo Rota, who has been said to be the founding father of modern algebraic combinatorics. (He is also my mathematical grandfather-to-be.)

Rota was a philosopher as well as a mathematician, and wrote an entire book primarily concerning the philosophy of mathematics. His book is called Indiscrete Thoughts.