# 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. – Peter Dukes Mar 24 '14 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. – Harry Huang Mar 26 '14 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? – Peter Dukes Mar 26 '14 at 7:50
• Well i have found it now, see my answer below @PeterDukes – Harry Huang Apr 4 '14 at 4:27