Journal of Integer Sequences, Vol. 19 (2016), Article 16.7.8 |

Department of Mathematics

University of Kentucky

Lexington, KY 40506-0027

USA

**Abstract:**

We give a new *q*-(1+*q*)-analogue of the
Gaussian coefficient, also known as the
*q*-binomial which, like the original *q*-binomial
,
is symmetric in *k* and *n*-*k*.
We show this *q*-(1+*q*)-binomial is more compact than
the one discovered by Fu, Reiner, Stanton, and Thiem.
Underlying our *q*-(1+*q*)-analogue
is a Boolean algebra decomposition
of an associated poset.
These ideas are extended to the Birkhoff transform of any finite poset.
We end with a discussion of higher analogues of the
*q*-binomial.

Received May 19 2016;
revised versions received August 31 2016; September 8 2016.
Published in *Journal of Integer Sequences*, September 11 2016.

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