Manfred Scheucher - cp's OEIS frontend

User: Manfred Scheucher

Manfred Scheucher has authored 43 sequences. Here are the ten most recent ones:

A368761 Number of labeled split graphs on n vertices such that {1..k} is independent and {k+1..n} is a clique for some k in {0..n}.

1, 2, 6, 24, 128, 928, 9280, 129152, 2515200, 68780544, 2647000064, 143580989440, 10988411686912, 1187350176604160, 181232621966082048, 39089521693818912768, 11916533065969825808384, 5135497592471003032846336, 3128995097443083790244380672, 2695613904312277811648715554816

Offset: 1

Robert Lauff and Manfred Scheucher, Jan 05 2024

nonn

Also the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple abc we have X(ab)X(ac)X(bc) not in {++-,+--}.

Cf. A048194.

seq(1 + add((2^k-1)*2^((n-1-k)*k),k=1..n-1),n=1..20); # Georg Fischer_, May 28 2024

def f(n): return 1+sum((2**k-1)*2**((n-1-k)*k) for k in range(1,n))

a(n) = 1 + Sum_{k=1..n-1} (2^k-1)*2^((n-1-k)*k).

a(20), a(21) joined by Georg Fischer, May 28 2024

A367448 Number of chordal graphs on n vertices with a fixed perfect elimination ordering (e.g., 1,2,3,...,n).

Original entry on oeis.org

1, 2, 7, 39, 324, 3839, 62973, 1402792, 41946319, 1673580047, 88922215948, 6297931501377, 596303138919753, 75787556639822258, 12991109500044250083, 3018313885461813882295, 955168488432838276254520, 413639698066068492610331231, 246197679553110860511406200613, 202212713843977008653180874488520

Offset: 1

Manfred Scheucher and Robert Lauff, Jan 05 2024

nonn

a(n) is the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple a

Cf. A048192.

a(n)={
  local(M=Map(Mat([1, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(proc(p,m)=for(k=0, poldegree(p), acc(p + x*(1 + x)^k, polcoef(p,k)*m)));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], proc(src[i, 1], src[i, 2])));
  vecsum(Mat(M)[,2])
} \\ Andrew Howroyd, Jan 06 2024

Terms a(12) and beyond from Andrew Howroyd, Jan 06 2024

A363586 a(n) is the number of mappings X:{{1..n} choose 3}->{+,-} such that X(a,b,c) = X(b,c,d) implies X(a,b,c) = X(a,b,d) = X(a,c,d) = X(b,c,d) for a < b < c < d.

Original entry on oeis.org

2, 10, 120, 3284, 199724, 26680618, 7782067930, 4936744315756

Offset: 3

Manfred Scheucher, Jun 13 2023

M. Balko, Ramsey numbers and monotone colorings, Journal of Combinatorial Theory, Series A, Volume 163 (2019), 34-58.

a(9)-a(10) from Bert Dobbelaere, Jun 26 2023

A361564 Number of (n-3)-connected unlabeled n-node graphs.

Original entry on oeis.org

4, 6, 10, 17, 25, 39, 59, 87, 127, 186

Offset: 3

Manfred Scheucher, Mar 15 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Cf. A259862.

a(n) = Sum_{j=1..3} A259862(n, n-j). - Andrew Howroyd, Mar 16 2023

A361578 Number of 5-connected polyhedra (or 5-connected simple planar graphs) with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 5, 8, 30, 85, 382, 1550, 7352

Offset: 12

Manfred Scheucher, Mar 16 2023

The icosahedral graph is the smallest 5-connected planar graph.

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Mathematics Stack Exchange -- What is the smallest 5-vertex-connected (5-edge-connected) planar graph?

Cf. A086216, A005470, A003094, A021103, A000944, A007027.

Cf. A049373 (planar graphs with minimum degree~5) and A111358 (5-connected planar trianguations)

A361367 Number of weakly 2-connected simple digraphs with n unlabeled nodes.

Original entry on oeis.org

7, 129, 7447, 1399245, 853468061, 1774125803324, 12983268697759210, 340896057593147232397, 32512334188761655225275067, 11365639780174824680535568799361, 14668665138188644335253106665956458513, 70315069858161131939222463684374769308619684

Offset: 3

Manfred Scheucher, Mar 09 2023

nonn

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A002218.

Cf. A000273, A003085.

\\ See links in A339645 for combinatorial species functions.
edges(v) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
{ my(N=15); Vec(-2*x^2 + OgfSeries(cycleIndexSeries(N))) } \\ Andrew Howroyd, Mar 09 2023

Terms a(7) and beyond from Andrew Howroyd, Mar 09 2023

A361366 Number of unlabeled simple planar digraphs with n nodes.

Original entry on oeis.org

1, 3, 16, 218, 9026, 907123

Offset: 1

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A005470.

Cf. A000273

A361368 Number of weakly connected simple planar digraphs with n unlabeled nodes.

Original entry on oeis.org

2, 13, 199, 8782, 897604

Offset: 2

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A003094.

Cf. A000273, A003085, A361366

A361369 Number of weakly 2-connected simple planar digraphs with n unlabeled nodes.

Original entry on oeis.org

7, 129, 6865, 774052

Offset: 3

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A021103.

Cf. A000273, A003085, A361366-A361368

A361371 Number of weakly 3-connected simple planar digraphs with n unlabeled nodes.

Original entry on oeis.org

42, 2688, 316208

Offset: 4

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A000944.

Cf. A000273, A003085, A361366-A361370

cp's OEIS Frontend

User: Manfred Scheucher

A368761 Number of labeled split graphs on n vertices such that {1..k} is independent and {k+1..n} is a clique for some k in {0..n}.

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A367448 Number of chordal graphs on n vertices with a fixed perfect elimination ordering (e.g., 1,2,3,...,n).

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A363586 a(n) is the number of mappings X:{{1..n} choose 3}->{+,-} such that X(a,b,c) = X(b,c,d) implies X(a,b,c) = X(a,b,d) = X(a,c,d) = X(b,c,d) for a < b < c < d.

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A361564 Number of (n-3)-connected unlabeled n-node graphs.

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A361578 Number of 5-connected polyhedra (or 5-connected simple planar graphs) with n nodes.

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A361367 Number of weakly 2-connected simple digraphs with n unlabeled nodes.

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PARI

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A361366 Number of unlabeled simple planar digraphs with n nodes.

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A361368 Number of weakly connected simple planar digraphs with n unlabeled nodes.

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A361369 Number of weakly 2-connected simple planar digraphs with n unlabeled nodes.

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A361371 Number of weakly 3-connected simple planar digraphs with n unlabeled nodes.

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