Robert Lauff - cp's OEIS frontend

User: Robert Lauff

Robert Lauff has authored 3 sequences.

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

A358471 a(n) is the number of transitive generalized signotopes.

Original entry on oeis.org

2, 14, 424, 58264, 33398288, 68779723376

Offset: 3

Robert Lauff, Nov 18 2022

A "transitive generalized signotope" is a generalized signotope X (cf. A328377) with the additional property that for any 5-tuple p, q, r, s, t, if (X(t,q,r), X(p,t,r), X(p,q,t), X(s,q,t), X(p,s,t), X(p,q,s)) = (+,+,+,+,+,+), then X(s,q,r)=+. Here X is extended to non-ordered triples by X(p(a),p(b),p(c)) = sgn(p)X(a,b,c) for any permutation p of three elements.
The "transitivity property" from the definition has a nice interpretation in the context of point sets, see "transitive interior triple systems" in Knuth.
The condition of transitivity from the definition above is implication (2.4a) in Knuth.
Every signotope (cf. A006247) is a transitive generalized signotope, giving a lower bound of 2^(c*n^2) <= a(n). This can be seen by checking the n=5 case. A violating 5-tuple in any signotope then cannot occur because it induces a signotope on 5 elements.

D. Knuth, Axioms and Hulls, Springer, 1992, 9-11.

Cf. A006247, A328377.

cp's OEIS Frontend

User: Robert Lauff

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}.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A358471 a(n) is the number of transitive generalized signotopes.

Views

Author

Keywords

Comments

References

Crossrefs