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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.

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  • $\begingroup$ This is now findstat.org/St001443. $\endgroup$
    – Martin Rubey
    Commented Jul 23, 2019 at 6:43
  • $\begingroup$ 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. $\endgroup$
    – Nathan Reading
    Commented Jul 23, 2019 at 15:35

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