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Statistic identifier: St001443

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Collection: Finite Cartan types

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Description: The largest coefficient in the Poincaré polynomial of the Weyl group of given Cartan type.

The Poincaré polynomial of a Weyl group $W$ is
$$
\sum_{w\in W} q^{\ell(w)} = \prod_i [d_i]_q,
$$
where $\ell$ denotes the Coxeter length, $d_1,\dots$ are the degrees (or exponents) of $W$ and $[n]_q=1 +\dots+q^{n-1}$ is the $q$-integer.

Thus, this statistic records the frequency of the most common length in the group.

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References: [1]   Gaichenkov, M. The growth of maximum elements for the reflection group $D_n$ [[MathOverflow:336756]]

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Code:
def statistic(C):
    from sage.combinat.q_analogues import q_int
    return max(prod(q_int(d, q) for d in WeylGroup(C).degrees()).list())


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Statistic values:

['A',1] => 1
['A',2] => 2
['B',2] => 2
['G',2] => 2
['A',3] => 6
['B',3] => 8
['C',3] => 8
['A',4] => 22
['B',4] => 46
['C',4] => 46
['D',4] => 30
['F',4] => 94
['A',5] => 101
['B',5] => 340
['C',5] => 340
['D',5] => 212
['A',6] => 573
['B',6] => 3210
['C',6] => 3210
['D',6] => 1924
['E',6] => 3662
['A',7] => 3836
['B',7] => 36336
['C',7] => 36336
['D',7] => 21280
['E',7] => 131046
['A',8] => 29228
['B',8] => 484636
['C',8] => 484636
['D',8] => 277788
['E',8] => 18210722

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Created: Jul 22, 2019 at 22:51 by Martin Rubey

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Last Updated: Aug 07, 2019 at 11:03 by Martin Rubey