# The number of partitions between two fixed partitions

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.

• Maybe this is only a minor technicality, but the upper bound on $i$ is throwing me off. Can you clarify? For example, between (2,1,1) and (2,2) are we allowing longer partitions such as (2,1,1,1) and/or larger partitions such as (2,2,1)? Or are there zero partitions in between these (presumably because of padding by zeros)? I would be happier if $l(M) \le l(N)$, but the way the upper bound on $i$ is written goes against this. Mar 24, 2014 at 8:44
• @PeterDukes Well, let's restrict our attention to what's called 'containment order', i.e. the containment of corresponding Ferrers diagram. (2,1,1) and (2,2) would not be comparable, under this order. Mar 26, 2014 at 5:00
• OK good. I am outside my comfort level here, but I agree this should be well known. Do you vaguely recall if the determinant was of some Hessenberg matrix? Mar 26, 2014 at 7:50
• Well i have found it now, see my answer below @PeterDukes Apr 4, 2014 at 4:27