Algorithm 1 DP based counting algorithm for $h(n,\mathsf{\Delta },m_{\le },d_{\le })$

Input: Integers $n-1\ge m\ge 0$ and $\mathsf{\Delta }\ge d\ge 0$.

Output:$h(n,\mathsf{\Delta },m_{\le },d_{\le })$.

$h[1,j,0_{=},0_{=}]:=h[1,j,0_{=},p_{\le }]:=h[1,j,0_{\le },p_{\le }]:=1$;

$h[i,j,0_{=},p_{\le }]:=h[i,j,0_{\le },p_{\le }]:=0$;

$h[2,j,1_{=},p_{=}]:=1;$ $h[2,j,1_{=},p_{\le }]:=h[2,j,1_{\le },p_{\le }]:=p+1$

for each $2\le i\le n,0\le j\le \mathsf{\Delta }$, $0\le p\le min\{j,d\}$;

for $i:=3,4,\dots ,n$ do

for $j:=0,1,\dots ,\mathsf{\Delta }$ do

for $k:=1,2,\dots ,min\{i,m\}$ do

for $p:=0,1,\dots ,min\{j,d\}$ do

if $p=0$ and $k=1$ then

$h[i,j,1_{=},0_{=}]:=h[i,j,1_{=},0_{\le }]:=h[i,j,1_{\le },0_{\le }]:=1$

else /* $p\ge 1$ or $k\ge 2$ */

$c:=1$; $h[i,j,k_{=},p_{=}]:=0;$ /* Initialization */

if $p=0$ then

$\ell :=\lfloor (i-1)/k\rfloor$

else /* $p\ge 1$ */

$\ell :=min\{\lfloor (i-1)/k\rfloor ,\lfloor j/p\rfloor \}$

end if;

for $q:=1,2,\dots ,\ell$ do

$c:=c·(h[k,p,k-1_{\le },p_{\le }]+q-1)/q;$

if $p=0$ then

$h[i,j,k_{=},p_{=}]:=h[i,j,k_{=},p_{=}]+c·h[i-qk,j,min{\{i-kq-1,k-1\}}_{\le },j_{\le }]$

else /* $p\ge 1$ */

$$\begin{gathered} h[i,j,k_{=},p_{=}]:=h[i,j,k_{=},p_{=}]+c·h[i-kq,j-pq,k_{=},min{\{j-pq,p- \\ 1\}}_{\le }]+h[i-kq,j-pq,min{\{i-kq-1,k-1\}}_{\le },j-p{q}_{\le }] \end{gathered}$$

end if

end for;

if $p=0$then /* $k\ge 2$ */

$h[i,j,k_{=},0_{\le }]:=h[i,j,k_{=},0_{=}]$

else /* $p\ge 1$ */

$h[i,j,k_{=},p_{\le }]:=h[i,j,k_{=},p-1_{\le }]+h[i,j,k_{=},p_{=}]$

end if;

$h[i,j,k_{\le },p_{\le }]:=h[i,j,k-1_{\le },p_{\le }]+h[i,j,k_{=},p_{\le }]$

end if

end for

end for

end for

end for;

output $h[n,\mathsf{\Delta },m_{\le },d_{\le }]$ as $h(n,\mathsf{\Delta },m_{\le },d_{\le })$.