A059441
Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1
Offset: 1
1;
1, 1;
1, 0, 1;
1, 3, 3, 1;
1, 0, 12, 0, 1;
1, 15, 70, 70, 15, 1;
1, 0, 465, 0, 465, 0, 1;
1, 105, 3507, 19355, 19355, 3507, 105, 1;
1, 0, 30016, 0, 1024380, ...;
1, 945, 286884, 11180820, 66462606, ...;
1, 0, 3026655, 0, 5188453830, ...;
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
- Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
- Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
- Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
- Wikipedia, Regular graph
-
Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)
-
for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019
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.
Original entry on oeis.org
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
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;
...
-
\\ 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
A333160
Number of non-isomorphic n X n symmetric binary matrices with an equal number of ones in every row and column up to permutation of rows and columns.
Original entry on oeis.org
1, 2, 3, 4, 6, 8, 17, 36, 146, 1066, 15419, 406684, 19284912, 1635874946, 249424764407, 68725494158824, 34418706513939926, 31487353344361957012, 52887877379630894268187, 163777247316556715401451972, 939121048579630147375554814224
Offset: 0
The a(2) = 3 matrices are:
[0 0] [1 0] [1 1]
[0 0] [0 1] [1 1]
A322704
Number of regular hypergraphs on n labeled vertices with no singletons.
Original entry on oeis.org
1, 1, 2, 4, 80, 209944
Offset: 0
The a(3) = 4 edge-sets:
{}
{{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
-
Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2^n-n-1}],{n,1,5}]
A333162
Number of regular graphs on n unlabeled nodes with half-edges.
Original entry on oeis.org
1, 2, 4, 6, 11, 16, 38, 78, 304, 1672, 19630, 450154, 20198620, 1673850034, 252574719718, 69216356716672, 34559361669877526, 31561018687380681802, 52958472442027295300520, 163901411216705405946164728, 939523329928455089556983977702
Offset: 0
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