A053418 -id:A053418 - cp's OEIS frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737

Offset: 0

Colin Mallows

nonn

Euler transform of A007719.
Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018
a(n) is the number of unlabeled loopless multigraphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

			a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):
  oo
  o o
  o-
  o -
  =
  --
  - -
From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12),
  (1)(1)(2)(2).
(End)
From _Gus Wiseman_, Jan 08 2024: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 rooted loopless multigraphs (root shown as singleton):
  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,2}}
         {{1},{2,3}}  {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{1,2},{3,4}}
                      {{1},{2,3},{2,3}}
                      {{1},{2,3},{2,4}}
                      {{1},{2,3},{4,5}}
(End)

Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Mateo Díaz, Dmitriy Drusvyatskiy, Jack Kendrick, and Rekha R. Thomas, Invariant Kernels: Rank Stabilization and Generalization Across Dimensions, arXiv:2502.01886 [math.OC], 2025. See p. 43.
Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67.

Row n=2 of A331485.

Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s=0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := RowSumMats[n, 2n, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

More terms from Vladeta Jovovic, Jan 26 2000

a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011

A000664 Number of graphs with n edges.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 68, 177, 497, 1476, 4613, 15216, 52944, 193367, 740226, 2960520, 12334829, 53394755, 239544624, 1111261697, 5320103252, 26237509076, 133087001869, 693339241737, 3705135967663, 20286965943329, 113694201046379, 651571521170323, 3815204365835840, 22806847476040913, 139088381010541237, 864777487052916454

Offset: 0

N. J. A. Sloane

These are simple graphs, unlabeled, with no isolated nodes, but are not necessarily connected.

			n=1: o-o (1)
n=2: o-o o-o, o-o-o (2)
n=3: o-o o-o o-o, o-o-o o-o, o-o-o-o, Y, triangle (5)
n=4: o-o o-o o-o o-o, o-o-o o-o o-o, o-o-o o-o-o, o-o o-o-o-o, o-o Y, o-o triangle,
o-o-o-o-o, >o-o-o, ><, square, triangle with tail (11)

W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 0..60
Nicolas Borie, The Hopf Algebra of graph invariants, arXiv preprint arXiv:1511.05843 [math.CO], 2015.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Fran Herr and Legrand Jones II, Iterated Jump Graphs, arXiv:2205.01796 [math.CO], 2022.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Overview of the 11 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 1
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 2
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 3
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 4
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 6
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 7
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 9
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11

Cf. A002905, A008406, A053418.
Row sums of A275421.
Cf. also A000088, A000055.

<< Combinatorica`; Table[NumberOfGraphs[2 n, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)
<< Combinatorica`; Table[Coefficient[GraphPolynomial[2 n, x], x, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)

a(n) = A008406(2*n,n). - Max Alekseyev, Sep 13 2016

Euler transform of A002905 (ignoring A002905(0)). - Franklin T. Adams-Watters Jul 03 2009

More terms from Vladeta Jovovic, Jan 08 2000, Aug 14 2007
Edited by N. J. A. Sloane, Feb 26 2008
Example for n=2 corrected by Adrian Falcone (falcone(AT)gmail.com), Jan 28 2009
Zeroth term inserted by Franklin T. Adams-Watters, Jul 03 2009
a(25)-a(26) from Max Alekseyev, Sep 19 2009
a(27)-a(60) from Max Alekseyev, Sep 07 2016

A053598 Number of n-node unlabeled digraphs without isolated nodes.

Original entry on oeis.org

1, 0, 2, 13, 202, 9390, 1531336, 880492496, 1792477159408, 13026163465206704, 341247403996148180800, 32522568124623933138617088, 11366712907916015518547782806784, 14669074325967499043636521641422216704, 70315641946149306808455637518883828774996992

Offset: 0

Vladeta Jovovic, Apr 10 2000

Equals first differences of A000273.

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A000273, A002494, A053418 (by # arcs). Row sums of A350908.

b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
      igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
    end:
a:= n-> b(n$2, [])-`if`(n=0, 0, b(n-1$2, [])):
seq(a(n), n=0..16);  # Alois P. Heinz, Sep 04 2019

Needs["Combinatorica`"];
nn=15;s=Sum[NumberOfDirectedGraphs[n]x^n,{n,0,nn}];CoefficientList[Series[s (1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 09 2012 *)
Join[{1}, Table[GraphPolynomial[n, x, Directed] /. x -> 1, {n, 0, 15}] // Differences] (* Jean-François Alcover, Feb 04 2015 *)

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053598(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

O.g.f.: A(x)*(1-x) where A(x) is o.g.f. for A000273. - Geoffrey Critzer, Oct 09 2012

A053454 Number of weakly connected digraphs with n arcs.

Original entry on oeis.org

1, 1, 4, 12, 53, 237, 1306, 7537, 47913, 322253, 2297874, 17191216, 134505656, 1095715055, 9267223594, 81162609328, 734511656413, 6856030049629, 65899370570285, 651338242941020, 6611459646337423, 68842439737228261

Offset: 0

Vladeta Jovovic, Jan 12 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column sums of A054733.
Row sums of A350789.
Cf. A003085, A053418, A350752.

\\ See A054733 for G, InvEulerMTS.
seq(n)=Vec(subst(Pol(InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))), x, 1)) \\ Andrew Howroyd, Jan 28 2022

Inverse Euler transform of A053418. - Max Alekseyev, Jan 22 2010

Extended by Max Alekseyev, Jan 22 2010

a(0)=1 prepended by Andrew Howroyd, Jan 28 2022

A121252 Number of labeled digraphs without isolated vertices and with n arcs.

Original entry on oeis.org

1, 2, 25, 520, 15150, 567540, 25986334, 1406214816, 87803248260, 6213408758960, 491423381371208, 42958461746823376, 4112928296979824348, 428022362222941505760, 48106759414197240877200, 5807382441785322296139392

Offset: 0

Vladeta Jovovic, Aug 22 2006

Cf. A054545, A053418 (unlabeled).

seq(value(Sum(binomial(m*(m-1),n)/2^(m+1),m=0..infinity)),n=0..20); # Vladeta Jovovic, Aug 25 2006

a(n) = Sum_{m>=0} binomial(m*(m-1),n)/2^(m+1). Column sums of A054547.

More terms from Max Alekseyev, Aug 23 2006

A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 3, 4, 4, 1, 1, 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 3, 34, 116, 331, 669, 1128, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 1, 15, 134, 664, 2535, 7796, 19719, 42193, 77324, 122960, 170317, 206983, 220768

Offset: 1

Andrew Howroyd, Jan 28 2022

			Triangle begins:
  [1] 0;
  [2] 0, 1, 1;
  [3] 0, 0, 3, 4,  4,  1,  1;
  [4] 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums are A053598.
Column sums are A053418.
The labeled version is A054547.
Cf. A052283, A054733.

\\ See A054733 for G.
row(n)={Vecrev(G(n,y)-G(n-1,y), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) }

cp's OEIS Frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

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A000664 Number of graphs with n edges.

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A053598 Number of n-node unlabeled digraphs without isolated nodes.

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A053454 Number of weakly connected digraphs with n arcs.

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A121252 Number of labeled digraphs without isolated vertices and with n arcs.

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A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

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