Combinatorial Interpretation of an Extension of Gaussian Polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:58:22Z http://mathoverflow.net/feeds/question/101473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101473/combinatorial-interpretation-of-an-extension-of-gaussian-polynomials Combinatorial Interpretation of an Extension of Gaussian Polynomials Ken Gonzales 2012-07-06T09:45:18Z 2012-07-06T10:08:15Z <p>It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over $GF(q)$.</p> <p>A generalization of $\binom{n}{k}_q$ are the so-called $p,q$-binomial coefficients, </p> <p>$\binom{n}{k}_{p,q}=\frac{{[n]}!}{[k]![n-k]!}$, </p> <p>where $[n]=\frac{p^n-q^n}{p-q}$ and $[n]!=[n][n-1]\cdots[2][1]$.</p> <p>The $p,q$-binomials equal the $q$-binomials when $q=1$.</p> <p>Question 1: Is there a vector space combinatorial interpretation for the $p,q$-binomials? If there is, how does the underlying two-parameter field look like? (There is a combinatorial interpretation in terms of tableaux and lattice paths but I'm more interested with the subspace interpretation.)</p> <p>Question 2 (somewhat related): A number of mathematicians have talked about the so-called $q$-disease (the widespread (at least those among working in $q$-series) interest in extending classical results to $q$-analogues.) Is there a $p,q$-disease?</p> <p>PS I'm supposed to write $[n]_{p,q}$ but it doesn't seem to work with \frac.</p>