Rich wrote:
I've been playing with Q-factorials:
QF(0) == 1 QF(N+1) == QF(N) * (1 + Q + Q^2 + ... + Q^N). [...] Q-binomials seem to be polynomials: QB(A,B) = QF(A+B) / (QF(A) QF(B)).
Other people have offered pointers and commented on q-Gamma, but the self-contained explanation of the Q-binomials is so nice that I can't let this go by. (I use this example as my introduction to q-deformations for grad students interested in the overlap of combinatorics and Lie theory.) The q-binomial is indeed a polynomial. The easy way to see that is to check the initially non-obvious recurrence relation, in your notation, QB(A,B) = QB(A-1,B) + Q^A QB(A,B-1) So what's really going on here? Well, think of the non-Q version, Binomial(a,b) = (a+b)!/a!b!, as counting the number of north/east grid walks from the southwest to northeast corners of an a-by-b rectangle. In that case, the usual recurrence relation B(a,b) = B(a-1,b) + B(a,b-1) reflects the fact that any such path begins with either a step north or east, after which it's a walk in a box with one side-length decreased. The Q version works exactly the same way, except that now, each walk contributes Q^w, where w is the number of unit boxes to the south/east of the walk. Some walks begin with a step east, in which case it's a normal walk in an (a-1)-by-b box. Others begin with a step north, in which case the entire bottom row of boxes are under the line, and the weight of the remaining walk in the a-by-(b-1) box should be increased by a (multiplied by Q^a). I can't stop without pointing out one last thing. In the non-Q version, a step north and a step east "commute", in that a walk containing ...NE... and the same with ...EN... get you to the same place and are otherwise indistinguishable. But in the Q version, these walks differ by a factor of Q, because one box has changed between being over and under the walk. This is a baby example of how the "generating function" Q is the same as the "quantum commutation" Q (where ab = Qba). --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.