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\\ Unlabeled Acyclic Digraphs
\\ Author: Andrew Howroyd, Dec 2021; updated Jan 2022
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\\ VERSION 1, Dec 2021
\\ This is fairly naive, see version 2 for something more efficient
\\ using inclusion-exclusion.
\\
\\ Essentially following R. W. Robinson's method.
\\ The labeled case is described in 1.6 of Harary and Palmer Graphical Enumeration.

\\ If intending to compute more than a few terms then give PARI more memory!
\\ This can be done with 'default'. For example:
\\ default(parisizemax, 10^9)


\\ Concatenates two sorted small vectors, preserving order.
\\ Used exclusively for combining two partitions.
\\ For example: Merge([1,1,4], [3,6]) => [1,1,3,4,6]
Merge(p1, p2)={
  my(v=vectorsmall(#p1+#p2), i=1, j=1, k=1);
  while(i<=#p1 && j<=#p2, if(p1[i]<=p2[j], v[k]=p1[i]; i++, v[k]=p2[j]; j++); k++);
  while(i<=#p1, v[k]=p1[i]; i++; k++);
  while(j<=#p2, v[k]=p2[j]; j++; k++); 
  v;
}

\\ Creates a map of partition to rank.
\\ The rank of the partition is the order it appears in the generation order of paritions.
\\ This is used to support matrices indexed by partition.
\\ Although there are more memory efficient ways of doing this taking advantage of
\\ the particular ordering of partiions, this is a small part of the overall memory footprint.
BuildPartitionIndex(n)={
   my(M=Map(), ix=1);
   forpart(p = n, mapput(M, p, ix); ix++);
   M;
}

\\ Gives the number of permutations of a particular type where the type is a partition.
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}


\\ This is the main function for building the enumeration of dags using layers.
\\ This is essentially the unlabeled equivalent of code given in A058876.
\\ The additional complexity comes from tracking counts by partition.
\\ Values instead of being simple integers are large matrices indexed by partitions.
\\ This could be done in a mathematically cleaner way using bivariate cycle index series.
\\ Hoevever such an implementation would likely require more program code and be less efficient.
\\ This is memory and cpu intensive.  
GenLayer(n, layers, pmap, y=1, z=1)={
  my(results = vector(n));
  for(i=1, n-1, \\ i is number of vertices in top shell.
    my(u=layers[n-i], M=matrix(numbpart(i), numbpart(#u)), pm=pmap[#u]);
    for(j=1, #u, \\ j is number of vertices that are top in previous shell.
       my(N=u[j], k=#u-j, Q=vector(max(j,k)), xi=0);
       forpart(pi=i, xi++;
         for(s=1, #Q, Q[s]=prod(t=1, #pi, my(g=gcd(s,pi[t])); (1+y^(s*pi[t]/g))^g) );
         my(b=z*binomial(n,i)*permcount(pi), xj=0);
         forpart(pj=j, xj++;
           my(w=b*prod(t=1, #pj, Q[pj[t]]-1), xk=0);
           forpart(pk=k, xk++;
              my(col=mapget(pm, Merge(pj,pk)));
              M[xi,col] += N[xj,xk]*w*prod(t=1, #pk, Q[pk[t]]);
           ); \\ pk
         ); \\ pj
       ); \\ pi
    ); \\ j
    results[i]=M;   
  ); \\ i
  my(M=matrix(numbpart(n), 1), xi=0);
  forpart(pi=n, xi++; M[xi,1]=permcount(pi));
  results[n]=M;
  results;
}

\\ Top level function
\\ y is an optional variable for counting by arcs (A350447).
\\ z is an optional variable for counting by length of longest directed path (A350448).
GenAllDag(n,y=1,z=1)={
  my(layers=vector(n), pmap=vector(n));
  for(i=1, n,
    pmap[i] = BuildPartitionIndex(i);
    layers[i] = GenLayer(i, layers, pmap, y, z);
  );
  layers;
}

\\ Helper for summing up values.
\\ w is an optional variable for number of sources (A122078)
SummarizeDagLayers(data, w=1)={
  my(results=vector(1+#data));
  results[1]=1;
  for(n=1, #data, 
    my(layer=data[n], s=0); 
    for(k=1, n,
      my(M=layer[k]);
      s += w^k*vecsum(sum(i=1, matsize(M)[1], M[i,]))/n!;
    );
    results[1+n] = s;
  );
  results;
}

\\ Triangle A122078: number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes.
AcyclicDigraphsByNonSources(n)={
  my(results=SummarizeDagLayers(GenAllDag(n), 'y));
  vector(n+1, n, Vec(results[n], n));
}

\\ Triangle A350447: number of acyclic digraphs on n unlabeled nodes with k arcs.
AcyclicDigraphsByArcsOld(n)={
  my(results=SummarizeDagLayers(GenAllDag(n, 'y)));
  [Vecrev(p) | p<-results];
}

\\ Triangle A350448: number of acyclic graphs on n unlabeled nodes whose longest directed path has k arcs.
AcyclicDigraphsByLongestPath(n)={
  my(results=SummarizeDagLayers(GenAllDag(n, 1, 'y)));
  vector(n+1, n, Vecrev(results[n], n));
}

\\ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\\ Version 2, Jan 2022
\\ This is an implementation of an answer by Mikhail Tikhomirov to a quetion on math-overflow.
\\ Link: https://mathoverflow.net/questions/395095/is-there-a-formula-for-the-number-of-st-dags-dag-with-1-source-and-1-sink-with
\\ References relating to this method: (hard to find on internet, so this list is thin)
\\ R. W. Robinson, Counting digraphs with restrictions on the strong components,
\\ Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
\\ See Eqn 18.

\\ acycic digraphs.
DagCycleIndexData(n, yf=e->2)={
  my(results=vector(n+1));
  results[1] = [1];
  for(n=1, n,
    my(v=vector(numbpart(n)), pm=BuildPartitionIndex(n));
    for(i=1, n,
      my(j=n-i, uj=results[j+1], b=binomial(n,i), Q=vector(i), xj=0);
      forpart(pj=j, xj++;
        for(s=1, #Q, Q[s]=prod(t=1, #pj, my(g=gcd(s, pj[t])); yf(s*pj[t]/g)^g));
        my(xi=0);
        forpart(pi=i, xi++;
           my(col=mapget(pm, Merge(pi,pj)));
           v[col] += uj[xj]*b*permcount(pi)*(-1)^(#pi-1)*prod(t=1, #pi, Q[pi[t]]);
        ); \\ pi
      ); \\ pj
    ); \\ i
    results[n+1] = v;
  ); \\ n
  results;
}

A350415seq(n)={my(v=DagCycleIndexData(n)); vector(n+1, n, vecsum(v[n])/(n-1)!)}
\\ Triangle A350447: number of acyclic digraphs on n unlabeled nodes with k arcs.
A350447rows(n)={my(v=DagCycleIndexData(n, e->1 + y^e)); [Vecrev(p) | p<-vector(n+1, n, vecsum(v[n])/(n-1)!)]}
AcyclicDigraphsByArcs(n)=A350447rows(n)

\\ Multivariate inverse Euler transform.
InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}

\\ Triangle A350449: number of weakly connected acyclic digraphs on n unlabeled nodes with k arcs.
WeakAcyclicDigraphsByArcs(n)={
  my(ci=DagCycleIndexData(n,e->1 + y^e));
  my(v=Vec(InvEulerMTS(Ser( vector(#ci, n, vecsum(ci[n])/(n-1)!) ))));
  [Vecrev(p) | p<-v];  
}

\\ Triangle A350450: number of unlabeled weakly connected acyclic digraphs with n arcs and k vertices.
WeakAcyclicDigraphsTr(n)={
  my(ci=DagCycleIndexData(n,e->1 + y^e + O(y^n)));
  my(s=InvEulerMTS(Ser( vector(#ci, n, vecsum(ci[n])/(n-1)!) )));
  vector(n, n, vector(n, k, polcoef(polcoef(s,k), n-1)));
}

\\ acycic digraphs with a global source.
DagWithGlobalSourceCIData(n, yf=e->2)={
  my(results=vector(n));
  results[1] = [1];
  for(n=2, n,
    my(v=vector(numbpart(n-1)), pm=BuildPartitionIndex(n-1));
    for(i=1, n-1,
      my(j=n-i-1, uj=results[j+1], b=binomial(n-1,i), Q=vector(i), xj=0);
      forpart(pj=j, xj++;
        for(s=1, #Q, Q[s]=yf(s)*prod(t=1, #pj, my(g=gcd(s, pj[t])); yf(s*pj[t]/g)^g) - 1);
        my(xi=0);
        forpart(pi=i, xi++;
           my(col=mapget(pm, Merge(pi,pj)));
           v[col] += uj[xj]*b*permcount(pi)*(-1)^(#pi-1)*prod(t=1, #pi, Q[pi[t]]);
        ); \\ pi
      ); \\ pj
    ); \\ i
    results[n] = v;
  ); \\ n
  results;
}

A350415seq(n)={my(v=DagWithGlobalSourceCIData(n)); vector(n, n, vecsum(v[n])/(n-1)!)}
A350488rows(n)={my(v=DagWithGlobalSourceCIData(n, e->1 + y^e)); [Vecrev(p) | p<-vector(n, n, vecsum(v[n])/(n-1)!)]}
A350490seq(n)={my(v=DagWithGlobalSourceCIData(n+1, e->1 + y^e + O(y*y^n))); Vec(vecsum(vector(#v, n, vecsum(v[n])/(n-1)!)))}

\\ acycic digraphs with a global source and sink.
DagWithGlobalSourceAndSinkCIData(n, yf=e->2)={
  my(source=DagWithGlobalSourceCIData(n-1, yf));
  my(results=vector(n));
  results[1] = [1];
  for(n=2, n,
    my(v=vector(numbpart(n-1)), pm=BuildPartitionIndex(n-1));
    for(i=1, n-1,
      my(j=n-i-1, uj=source[j+1], b=binomial(n-1,i), Q=vector(i), xj=0);
      forpart(pj=j, xj++;
        for(s=1, #Q, Q[s]=yf(s)*prod(t=1, #pj, my(g=gcd(s, pj[t])); yf(s*pj[t]/g)^gcd(s, pj[t])) - 1);
        my(xi=0);
        forpart(pi=i, xi++;
           my(col=mapget(pm, Merge(pi,pj)), w=sum(t=1, #pi, pi[t]==1));
           v[col] += uj[xj]*b*w*permcount(pi)*(-1)^(#pi-1)*prod(t=1, #pi, Q[pi[t]]);
        ); \\ pi
      ); \\ pj
    ); \\ i
    results[n] = v;
  ); \\ n
  results;
}


A345258seq(n)={my(v=DagWithGlobalSourceAndSinkCIData(n)); vector(#v, n, vecsum(v[n])/(n-1)!)}
A350491rows(n)={my(v=DagWithGlobalSourceAndSinkCIData(n, e->1 + y^e)); [Vecrev(p) | p<-vector(n, n, vecsum(v[n])/(n-1)!)]}
A350492seq(n)={my(v=DagWithGlobalSourceAndSinkCIData(n+1, e->1 + y^e + O(y*y^n))); Vec(vecsum(vector(#v, n, vecsum(v[n])/(n-1)!)))}

\\ End