A333157 Triangle read by rows: T(n,k) is the number of n X n symmetric binary matrices with k ones in every row and column.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 18, 10, 1, 1, 26, 112, 112, 26, 1, 1, 76, 820, 1760, 820, 76, 1, 1, 232, 6912, 35150, 35150, 6912, 232, 1, 1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1, 1, 2620, 708256, 24243520, 133948836, 133948836, 24243520, 708256, 2620, 1

Offset: 0

Author

Andrew Howroyd, Mar 09 2020

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of k-regular symmetric relations on n labeled nodes.

T(n,k) is the number of k-regular graphs with half-edges on n labeled vertices.

Terms may be computed without generating all graphs by enumerating the number of graphs by degree sequence. A PARI program showing this technique is given below. Burnside's lemma as applied in A122082 and A000666 can be used to extend this method to the case of unlabeled vertices A333159 and A333161 respectively.

Examples

			Triangle begins:
  1,
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,  10,    18,     10,       1;
  1,  26,   112,    112,      26,      1;
  1,  76,   820,   1760,     820,     76,     1;
  1, 232,  6912,  35150,   35150,   6912,   232,   1;
  1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..230

Crossrefs

Columns k=0..8 are A000012, A000085, A000986, A110040, A139670, A139671, A139673, A139674, A139675.

Row sums are A322698.

Central coefficients are A333164.

Cf. A188448 (transposed as array).

Cf. A059441, A295193, A333158, A333159, A333161.

Programs

PARI

\\ See script in A295193 for comments.
GraphsByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([x^0,1])));
  my(acc(p,v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r,p,i,q,v,e) = if(e<=limit && poldegree(q)<=limit, if(i<0, if(ok(x^e+q, r), acc(x^e+q, v)), my(t=polcoeff(p,i)); for(k=0,t,self()(r,p,i-1,(t-k+x*k)*x^i+q,binomial(t,k)*v,e+k)))));
  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i,1]); recurse(n-k, p, poldegree(p), 0, src[i,2], 0))); Mat(M);
}
Row(n)={my(M=GraphsByDegreeSeq(n, n\2, (p,r)->poldegree(p)-valuation(p,x) <= r + 1), v=vector(n+1)); for(i=1, matsize(M)[1], my(p=M[i,1], d=poldegree(p)); v[1+d]+=M[i,2]; if(pollead(p)==n, v[2+d]+=M[i,2])); for(i=1, #v\2, v[#v+1-i]=v[i]); v}
for(n=0, 8, print(Row(n))) \\ Andrew Howroyd, Mar 14 2020

Formula

T(n,k) = T(n,n-k).