A049312 Number of graphs with a distinguished bipartite block, by number of vertices.

1, 2, 4, 8, 17, 38, 94, 258, 815, 3038, 13804, 78760, 580456, 5647602, 73645352, 1297920850, 31031370360, 1007551636038, 44432872400460, 2661065508648436, 216457998880015366, 23920728651724212120, 3593384834863975164882, 734240676501745813835934

Offset

0,2

Comments

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, apply EULER transform.

Inverse Euler transform is A318870.

References

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..40

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Karen L. Collins, Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.

M. Guay-Paquet, A. H. Morales and E. Rowland, Structure and enumeration of (3+ 1)-free posets, arXiv preprint arXiv:1212.5356 [math.CO], 2012-2013. - From N. J. A. Sloane, Feb 01 2013

J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.

J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.

E. M. Wright, The k-connectedness of bipartite graphs, J. Lond. Math. Soc. (2), 25 (1982), 7-12.

Formula

a(n) ~ 1/n! A047863(n) = 1/n! Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see Wright; see also Thm. 3.7 of the Troyka link, which cites Wright). - Justin M. Troyka, Oct 29 2018

Example

a(2)=4: null graph with 0, 1 or 2 vertices in the distinguished block and complete graph with 1 vertex in distinguished block.

Maple

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
a:= d-> add(A(n, d-n), n=0..d):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 01 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
a[d_] := Sum[A[n, d-n], {n, 0, d}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

Crossrefs

Row sums of A028657.

Cf. A048194, A318870.

Sequence in context: A101516 A118928 A325921 * A132043 A055545 A241671

Adjacent sequences: A049309 A049310 A049311 * A049313 A049314 A049315

Keyword

nonn,nice

Author

Peter J. Cameron

Extensions

More terms from Vladeta Jovovic, Jun 17 2000

Status

approved