BlockMatrix[{{GraphAdjacencyMatrix[HaagerupBigraph],-IdentityMatrix[GraphRank[HaagerupBigraph]]},{IdentityMatrix[GraphRank[HaagerupBigraph]],ZeroMatrix[GraphRank[HaagerupBigraph]]}}]//MatrixForm

HomsToTrivial[g_GradedBigraph,n_]:=MatrixPower[BlockMatrix[{{GraphAdjacencyMatrix[g],-IdentityMatrix[GraphRank[g]]},{IdentityMatrix[GraphRank[g]],ZeroMatrix[GraphRank[g]]}}],n][[1,1]]

Table[HomsToTrivial[HaagerupBigraph,k],{k,1,30}]

Table[HomsToTrivial[DualHaagerupBigraph,k],{k,1,30}]

Table[HomsToTrivial[haagerupFamilyBigraph[7],k],{k,15,38}]

Table[HomsToTrivial[haagerupFamilyBigraph[11],k],{k,23,46}]

PartitionsP[3]

TrainsVsHoms[g_GradedBigraph,d_]:={HomsToTrivial[g,2(DepthOfBranchPoint[g]+1)+2d],}

DisplayBigraph[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}]]

Table[HomsToTrivial[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],k],{k,6,14,2}]

Table[HomsToTrivial[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],k],{k,10,18,2}]

Table[DimensionOfLowWeightSpace[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}]][k],{k,5,9}]

Table[HomsToTrivial[HaagerupAsaedaBigraph,k],{k,10,20,2}]