Denote by $I_m=\{0,1,2,…m\}$, by $N_s=\{1,2,…,s\}$ , by $\overline s$ least common multiple of elements of set $N_s$ and by $p(k,N_s)$ the number of partitions of natural number $k$ in parts used from set $N_s $
I can prove that
$p(\overline {s}n+l,N_s)=p(l,N_s)+\sum_{i\in I_{\frac{\overline s}{s}}}\sum_{j\in I_n} p(\overline {s}n+l-\overline {s}j-si,N_{s-1})$
For $ s>0$ define a function
$A_{n}^{s}(a,r)=\sum_{j\in I_n} p(\overline {s}\left(\frac{\overline {s+1}}{\overline s}n-\frac{\overline {s+1}}{\overline s} j+a\right)+r,N_s),a\in Z,r\in I_{\overline s}$
$A_{n}^{s}(a,r)=0,a\notin Z$
Still there I am all right, but problem for me is how under these conditions get the equation
$ p(\overline {s}n+l,N_s)- p(l,N_s)= \sum_{r\in I_{\overline {s-1}}}\sum_{i\in I_{\frac{\overline s}{s}}}A_{n}^{s-1}\left(\frac{l-si-r}{\overline {s-1}},r\right)$
I am sure that this formula is correct but some details of proof remain unclear for me.