# Typo in Stanley, Enumerative combinatorics II, Cor. 7.23.9?

In Stanley, EC2, we have the following statement:

I think there is a typo in the first sum after "generating function", and that $$[n]_q!$$ should be replaced by $$(1-q)(1-q^2)\dotsb (1-q^n)$$, and same for $$[n]_t!$$.

The problem with the proof seems to be the line

which is incorrect(?) as the denominator on the right hand side should be not the q/t-factorials, but the above product, according to the lemma which is referenced there.

This issue (if my suspicions hold), is not listed in the Errata..

I have also checked this in Mathematica, so I am fairly confident this is a typo.

• Looks like it. It's easy to mix up $[n]_q!$ and $(1-q)(1-q^2)\cdots(1-q^n)$ (which of course differ only by a factor of $(1-q)^n$): that's a mistake that I make all the time when writing these kind of generating functions. Commented Feb 17, 2021 at 14:33
• @SamHopkins yes, exactly my thought... Commented Feb 17, 2021 at 15:11
• Thanks, I will fix this. Commented Feb 17, 2021 at 19:42
• On second thought, just before Corollary 7.23.9 I define $[n]!_q=(1-q)(1-q^2)\cdots (1-q^n)$, and similarly for $[n]!_t$. My notation for $[n]!_q/(1-q)^n$ is $(n)!$ (in boldface---MO displays all math in boldface, which doesn't work so well for my notation). Commented Feb 18, 2021 at 2:51
• @RichardStanley Ah, I see! My mistake for not looking up the convention! Commented Feb 18, 2021 at 7:41

The (somewhat nonstandard) convention in the text, stated just before the corollary, is that $$[n]!_q = (1-q)(1-q^2)\cdots(1-q^n)$$. For the quantity $$[n]!_q/(1-q)^n$$, the notation $$\mathbf{(n)!}$$ is used instead.