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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 forms the rank of the vector space with the well known generating function for inversions.

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.

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 forms the rank of the vector space with the well known generating function for inversions.

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 forms the rank of the vector space with the well known generating function for inversions.