A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Author

Vladeta Jovovic, Dec 29 1999

Keywords

• nonn

Comments

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017

Row sums of the triangle defined by the Multiset Transformation of A076864,

1 ;

0 1;

0 2 1;

0 5 2 1;

0 12 8 2 1;

0 33 22 8 2 1;

0 103 72 26 8 2 1;

0 333 229 87 26 8 2 1;

0 1183 782 295 92 26 8 2 1;

0 4442 2760 1036 315 92 26 8 2 1;

0 17576 10270 3735 1129 321 92 26 8 2 1;

0 72810 39770 13976 4117 1154 321 92 26 8 2 1;

0 314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;

- R. J. Mathar, Jul 18 2017

Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

Examples

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)
		

References

• Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
• Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
• Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
• Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
• Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Crossrefs

Cf. A001399, A003082, A014395, A014396, A014397, A014398.

Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.

Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

Programs

Mathematica

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

Formula

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

Extensions

More terms from Sean A. Irvine, Oct 02 2011