TITLE: The number of partitions between two fixed partitions
QUESTION [5 upvotes]: Given two partitions M and N, with $M_i \leq N_i$ for all $1\leq i\leq  \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum_{\lambda:  M_i\leq \lambda_i\leq  N_i} q^{|\lambda|}$$ 
I remember seeing a determinant formula before, the elements of the determinant being q-binomial coefficients. But now I can't find it.

REPLY [2 votes]: Note on enumeration of partitions contained in a given shape
by Ira M. Gessel and Nicholas Loehr
link:
http://www.sciencedirect.com/science/article/pii/S0024379509004819