Geodesic
The Gaussian coefficient revisited
Journal of integer sequences, Tome 19 (2016) no. 7
We give a new $q-(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\genfrac{[}{]}{0pt}{}{n}{k}_q$, 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.
Classification : 06A07, 05A05, 05A10, 05A30
Keywords: q-analogue, Birkhoff transform, distributive lattice, poset decomposition 
@article{JIS_2016__19_7_a6,
     author = {Ehrenborg,  Richard and Readdy,  Margaret A.},
     title = {The {Gaussian} coefficient revisited},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {7},
     zbl = {1352.05036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_7_a6/}
}
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Ehrenborg,  Richard; Readdy,  Margaret A. The Gaussian coefficient revisited. Journal of integer sequences, Tome 19 (2016) no. 7. http://geodesic.mathdoc.fr/item/JIS_2016__19_7_a6/