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

Author

Peter J. Cameron

Keywords

• nonn
• nice

Comments

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

Inverse Euler transform is A318870.

Examples

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

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.

Crossrefs

Row sums of A028657.

Cf. A048194, A318870.

Programs

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 *)

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

Extensions

More terms from Vladeta Jovovic, Jun 17 2000