Pensieve header: Counting virtual pure braids.

CountVPB[n_,m_]:=Module[{gens,diags,s,rels,r=0,ij,ijk,ijkl,i,j,k,l,mat},
gens=Flatten@Table[{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), i,j])},{i,n},{j,DeleteCases[Range@n,i]}];
diags=Flatten[Table[VPB[n,#]&/@Tuples[gens,s], {s,0,m}]];
rels=Flatten[{
Table[{i,j}=ij; {
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), i,j])},q]],VPB[n,Join[p,q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), i,j]),(Subscript[σ, i,j])},q]],VPB[n,Join[p,q]]]
},
{s,0,m-2},{t,0,s},{p,Tuples[gens,t]},{q,Tuples[gens,s-t]},
{ij,Join@@(Permutations/@Subsets[Range[n],{2}])}
],
Table[{i,j,k}=ijk; {
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[σ, i,k]),(Subscript[σ, j,k])},q]],VPB[n,Join[p,{(Subscript[σ, j,k]),(Subscript[σ, i,k]),(Subscript[σ, i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), j,i]),(Subscript[σ, i,k]),(Subscript[σ, j,k])},q]],VPB[n,Join[p,{(Subscript[σ, j,k]),(Subscript[σ, i,k]),(Subscript[(OverBar[σ]), j,i])},q]]],
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[σ, i,k]),(Subscript[(OverBar[σ]), k,j])},q]],VPB[n,Join[p,{(Subscript[(OverBar[σ]), k,j]),(Subscript[σ, i,k]),(Subscript[σ, i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[σ, i,j]),(Subscript[(OverBar[σ]), k,i]),(Subscript[(OverBar[σ]), k,j])},q]],VPB[n,Join[p,{(Subscript[(OverBar[σ]), k,j]),(Subscript[(OverBar[σ]), k,i]),(Subscript[σ, i,j])},q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), j,i]),(Subscript[(OverBar[σ]), k,i]),(Subscript[σ, j,k])},q]],VPB[n,Join[p,{(Subscript[σ, j,k]),(Subscript[(OverBar[σ]), k,i]),(Subscript[(OverBar[σ]), j,i])},q]]],
T[VPB[n,Join[p,{(Subscript[(OverBar[σ]), j,i]),(Subscript[(OverBar[σ]), k,i]),(Subscript[(OverBar[σ]), k,j])},q]],VPB[n,Join[p,{(Subscript[(OverBar[σ]), k,j]),(Subscript[(OverBar[σ]), k,i]),(Subscript[(OverBar[σ]), j,i])},q]]]
},
{s,0,m-3},{t,0,s},{p,Tuples[gens,t]},{q,Tuples[gens,s-t]},
{ijk,Join@@(Permutations/@Subsets[Range[n],{3}])}
]
}];
mat=SparseArray[
Flatten@Cases[rels,T[b1_,b2_]:>{{++r,Position[diags,b1][[1,1]]}->1, {r,Position[diags,b2][[1,1]]}->-1}],
{r,Length@diags}
];
Length@diags-If[r==0,0,MatrixRank@mat]
]

CountVPB[2,1]

CountVPB[2,2]

CountVPB[2,3]

CountVPB[2,4]

CountVPB[3,1]

CountVPB[3,2]

CountVPB[3,3]

CountVPB[3,4]

CountVPB[3,5]