SCountOUForms[n_,m_]:=Module[{gens},
gens=Flatten@Table[{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), i,j])},{i,n},{j,DeleteCases[Range@n,i]}];
Length@Union@Flatten@Table[R2Reduce@UOFix@VPB[n,Sequence@@p],{p,Tuples[gens,m]}]]

SCountOUForms[4,2]

b42=Flatten@{
VPB[4],
Table[VPB[4,(Subscript[s, t,h])],
{t,4},{h,Range[4]~Complement~{t}},{s,{σ,(OverBar[σ])}}
],
Table[{t2,h2}={t1,h1}[[o2]]; VPB[4,(Subscript[s1, t1,h1]),(Subscript[s2, t2,h2])],
{t1,4},{h1,Range[4]~Complement~{t1}},{s1,{σ,(OverBar[σ])}},{o2,{{1,2},{2,1}}},{s2,{σ,(OverBar[σ])}}
],
Table[{t1,h1}={t,h}[[o1]];{t2,h2}={{t,h}[[w1]],Complement[Range[4],{t,h}][[w2]]}[[o2]]; VPB[4,(Subscript[s1, t1,h1]),(Subscript[s2, t2,h2])],
{t,4},{h,Range[4]~Complement~{t}},{s1,{σ,(OverBar[σ])}},{o1,{{1,2},{2,1}}},{s2,{σ,(OverBar[σ])}},{o2,{{1,2},{2,1}}},{w1,2},{w2,2}
],
Table[{t1,h1}={1,e1}[[o1]]; {t2,h2}=DeleteCases[{2,3,4},e1][[o2]];VPB[4,(Subscript[s1, t1,h1]),(Subscript[s2, t2,h2])],
{e1,2,4},{s1,{σ,(OverBar[σ])}},{o1,{{1,2},{2,1}}},{s2,{σ,(OverBar[σ])}},{o2,{{1,2},{2,1}}}
]
}

v42=R2Reduce/@UOFix/@b42

Length@Union@v42

With[{m2=2,n=4},
gens=Flatten@Table[{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), i,j])},{i,n},{j,DeleteCases[Range@n,i]}];
Table[{i,j,k,l}=ijkl[[perm]]; {
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[σ, k,l])},q]],VPB[n,Join[p,{(Subscript[σ, k,l]),(Subscript[σ, i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), i,j]),(Subscript[σ, k,l])},q]],VPB[n,Join[p,{(Subscript[σ, k,l]),(Subscript[(OverBar[σ]), i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), k,l])},q]],VPB[n,Join[p,{(Subscript[(OverBar[σ]), k,l]),(Subscript[σ, i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), i,j]),(Subscript[(OverBar[σ]), k,l])},q]],VPB[n,Join[p,{(Subscript[(OverBar[σ]), k,l]),(Subscript[(OverBar[σ]), i,j])},q]]]
},
{s,0,m2-2},{t,0,s},{p,Tuples[gens,t]},{q,Tuples[gens,s-t]},
{ijkl,Subsets[Range[n],{4}]},{perm,{{1,2,3,4},{1,3,2,4},{1,4,2,3}}}
]]