# The growth of maximum elements for the reflection group $D_n$

Let's consider maximum coefficients in Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) $$D_n$$ as mentioned in A162206. The maximal numbers $$M(n)$$: $$1,2,6,30,212,1924,21280,...$$

$$\frac {6} {2} = 3$$;

$$\frac {30} {6} =5$$;

$$\frac {212} {30}=7,06666667$$;

$$\frac {1924} {212}=9,0754717$$;

$$\frac {21280} {1924}=11,0602911$$

How to prove $$\frac {M_{n+1}} {M_{n}} \approx 2n-1$$ for large $$n$$?

I know that the similar ratio for row max of Mahonian numbers $$T(n,k)$$ yields $$\approx n-\frac {1} {2}$$ while the numbers form the rank of the vector space with the well known generating function for inversions.

• This is now findstat.org/St001443. Jul 23 '19 at 6:43
• I haven't thought this through to the end, but perhaps there is an argument based on writing the Poincare polynomials in terms of the degrees. It is certainly easy to write a recursion giving the actual polynomials. From there, maybe one can see how largest coefficients behave under the recursion. Have you tried that? UPDATE: Based on the OEIS sequence you reference, I'm now guessing you've already thought about this. Jul 23 '19 at 15:35