[seqfan] A possible interpretation of A084239 in terms of the q-binomial coefficients
Vladimir Reshetnikov
v.reshetnikov at gmail.com
Sat Nov 12 03:43:22 CET 2016
Dear SeqFans,
I was studying properties of the q-binomial coefficients (also known as the
Gaussian binomial coefficients):
http://mathworld.wolfram.com/q-BinomialCoefficient.html,
http://en.wikipedia.org/wiki/Gaussian_binomial_coefficient. Recall that,
like the regular binomial coefficients, the q-binomial coefficients have
two integer parameters n, k, but also have an additional complex (or real)
parameter q. For q = 1 they reduce to the regular binomial coefficients,
but in general they are polynomials in q with some positive integer
coefficients. These polynomials can be arranged in a triangular table
similar to the Pascal triangle of the regular binomial coefficients. It
begin as follows:
1
1, 1
1, 1 + q, 1
1, 1 + q + q^2, 1 + q + q^2, 1
1, 1 + q + q^2 + q^3, 1 + q + 2 q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1
...
There are several sequences in the OEIS related to the q-binomial
coefficients. For the sake of experiment, I decided to replace each of the
polynomials in the table with its largest coefficient. It gives the
following triangle:
1
1, 1
1, 1, 1
1, 1, 1, 1
1, 1, 2, 1, 1
1, 1, 2, 2, 1, 1
1, 1, 3, 3, 3, 1, 1
...
Finally, let us sum all numbers in each row of this table. We get the
sequence:
1, 2, 3, 4, 6, 8, 13, 20, 32, 52, 90, 152, 268, 472, ...
A lookup in the OEIS returns a possible match (all 36 terms given there
match exactly) http://oeis.org/A084239, whose name is "Rank of K-groups of
Furstenberg transformation group C*-algebras of n-torus.". Now, I have not
a slightest idea what the Furstenberg transformation is, so I am unable to
check if it is really the same sequence, but comments in the entry mention
some connections to other combinatorial sequences. So, I am asking for help
from anybody familiar with this area, and for an advice about in what form
it would be better to add this information to the OEIS.
Here is the *Mathematica* program that I used to compute this sequence:
Table[Sum[Max[CoefficientList[FunctionExpand[QBinomial[n, k, q]], q]], {k,
0, n}], {n, 0, 35}]
--
Thanks
Vladimir
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