A051031 -id:A051031 - cp's OEIS frontend

A008483 Number of partitions of n into parts >= 3.

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011
Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011
By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.
For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005
(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009
Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009
Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014
For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482.  For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014
From Peter Bala, Dec 01 2024: (Start)
Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)
Peter Adams, Saad I. El-Zanati, Peter Florido, and William Turner, On 2-Factorizations of the Complete 3-Uniform Hypergraph of Order 12 Minus a 1-Factor, Combinatorics, Graph Theory and Computing (SEICCGTC 2021) Springer Proc. Math. Stat., Vol 448, pp. 383-392. See p. 326.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
R.-Q. Feng, J. H. Kwak and E. K. Lloyd, Isomorphism classes of authentication codes, Bull. Austral. Math. Soc. 69 (2004), no. 2, 203-215.
Elisabeth Gaar and Daniel Krenn, Metamour-regular Polyamorous Relationships and Graphs, arXiv:2005.14121 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446
F. Jouneau-Sion and O. Torres, In Fisher's net: exact F-tests in semi-parametric models with exchangeable errors, August 2014, preprint on ResearchGate.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g.
Johan Kok, Degree affinity number of certain 2-regular graphs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
Eric Weisstein's World of Mathematics, Two-Regular Graph.

Essentially the same sequence as A026796 and A281356.
From Jason Kimberley, Nov 07 2009 and Jan 05 2011 and Feb 03 2011: (Start)
Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), this sequence (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected).
2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), this sequence (g=3), A008484 (g=4), A185325 (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10), ... (End)
Cf. A008284.

p := NumberOfPartitions; A008483 :=  func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; // Jason Kimberley, Jan 11 2011

series(1/product((1-x^i),i=3..50),x,51);
ZL := [ B,{B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); # Zerinvary Lajos, Mar 13 2007
with(combstruct):ZL2:=[S,{S=Set(Cycle(Z,card>2))}, unlabeled]:seq(count(ZL2,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
with(combstruct):a:=proc(m) [A,{A=Set(Cycle(Z,card>m))},unlabeled]; end: A008483:=a(2):seq(count(A008483,size=n),n=0..46); # Zerinvary Lajos, Oct 02 2007

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *)
Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]]  (* Clark Kimberling, Feb 27 2014 *)

a(n) = numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import partition
def A008483(n): return partition(n)-partition(n-1)-partition(n-2)+partition(n-3) # Chai Wah Wu, Jun 10 2025

a(n) = p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts (A000041).
G.f.: Product_{m>=3} 1/(1-x^m).
G.f.: (Sum_{n>=0} x^(3*n)) / (Product_{k=1..n} (1 - x^k)). - Joerg Arndt, Apr 17 2011
a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller, Aug 14 2006
Euler transformation of A179184. a(n) = A179184(n) + A165652(n). - Jason Kimberley, Jan 05 2011
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015
G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
a(n) = Sum_{j=0..floor(n/2)} A008284(n-2*j,j). - Gregory L. Simay, Apr 27 2023
G.f.: 1 + Sum_{n >= 1} x^(n+2)/Product_{k = 0..n-1} (1 - x^(k+3)). - Peter Bala, Dec 01 2024

A005176 Number of regular graphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, 546, 19002, 389454, 50314870, 2942198546, 1698517037030, 442786966117636, 649978211591622812, 429712868499646587714, 2886054228478618215888598, 8835589045148342277802657274, 152929279364927228928025482936226, 1207932509391069805495173417972533120, 99162609848561525198669168653641835566774

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. Friedman, Illustration of small graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Jennifer M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Regular Graph.

Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Simple regular graphs of any degree: A005177 (connected), A068932 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs with girth at least g: this sequence (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).
Cf. A295193.

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002

Row sums of triangle A051031.

More terms from David Wasserman, Mar 08 2002
a(15) and a(16) from Jason Kimberley, Sep 25 2009
Edited by Jason Kimberley, Jan 06 2011 and May 24 2012
a(17)-a(21) from Andrew Howroyd, Mar 08 2020
a(22)-a(24) from Andrew Howroyd, Apr 05 2020

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A068934 Triangular array C(n, r) = number of connected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 5, 6, 3, 1, 1, 0, 0, 1, 0, 16, 0, 4, 0, 1, 0, 0, 1, 19, 59, 60, 21, 5, 1, 1, 0, 0, 1, 0, 265, 0, 266, 0, 6, 0, 1, 0, 0, 1, 85, 1544, 7848, 7849, 1547, 94, 9, 1, 1, 0, 0, 1, 0, 10778, 0, 367860, 0

Offset: 1

David Wasserman, Mar 08 2002

A graph is called r-regular if every node has exactly r edges. The numbers in this table were copied from the column sequences.

This sequence can be derived from A051031 by inverse Euler transform. See the comments in A051031 for a brief description of how that sequence can be computed without generating all regular graphs. - Andrew Howroyd, Mar 13 2020

			01: 1;
02: 0, 1;
03: 0, 0, 1;
04: 0, 0, 1, 1;
05: 0, 0, 1, 0, 1;
06: 0, 0, 1, 2, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 5, 6, 3, 1, 1;
09: 0, 0, 1, 0, 16, 0, 4, 0, 1;
10: 0, 0, 1, 19, 59, 60, 21, 5, 1, 1;
11: 0, 0, 1, 0, 265, 0, 266, 0, 6, 0, 1;
12: 0, 0, 1, 85, 1544, 7848, 7849, 1547, 94, 9, 1, 1;
13: 0, 0, 1, 0, 10778, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 1, 509, 88168, 3459383, 21609300, 21609301, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 1, 0, 805491, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 1, 4060, 8037418, 2585136675, 113314233808, 733351105934, 733351105935, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24, first 16 rows from Jason Kimberley)
Jason Kimberley, Connected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

Connected regular simple graphs: A005177 (any degree -- sum of rows), this sequence (triangular array), specified degree r (columns): A002851 (r=3), A006820 (r=4), A006821 (r=5), A006822 (r=6), A014377 (r=7), A014378 (r=8), A014381 (r=9), A014382 (r=10), A014384 (r=11).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: this sequence (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: A186733 (g=3), A186734 (g=4).

C(n, r) = A051031(n, r) - A068933(n, r).

Column k is the inverse Euler transform of column k of A051031. - Andrew Howroyd, Mar 10 2020

Edited by Jason Kimberley, Sep 23 2009, Nov 2011, Jan 2012, and Mar 2012

A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1

Offset: 1

N. J. A. Sloane, Feb 01 2001

			1;
1,   1;
1,   0,       1;
1,   3,       3,        1;
1,   0,      12,        0,          1;
1,  15,      70,       70,         15,    1;
1,   0,     465,        0,        465,    0,   1;
1, 105,    3507,    19355,      19355, 3507, 105, 1;
1,   0,   30016,        0,    1024380, ...;
1, 945,  286884, 11180820,   66462606, ...;
1,   0, 3026655,        0, 5188453830, ...;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
Wikipedia, Regular graph

Row sums are A295193.
Columns: A123023 (k=1), A001205 (k=2), A002829 (k=3, with alternating zeros), A005815 (k=4), A338978 (k=5, with alternating zeros), A339847 (k=6).
Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.

Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)

for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(37)-a(55) from Andrew Howroyd, Aug 25 2017

A014378 Number of connected regular graphs of degree 8 with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848873, 214755319657939505395, 187549729101764460261498, 186685399408147545744203815, 210977245260028917322933154987

Offset: 0

N. J. A. Sloane

Since the nontrivial 8-regular graph with the least number of vertices is K_9, there are no disconnected 8-regular graphs with less than 18 vertices. Thus for n<18 this sequence is identical to A180260. - Jason Kimberley, Sep 25 2009 and Feb 10 2011

			a(0)=1 because the null graph (with no vertices) is vacuously 8-regular and connected.

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Octic Graph
Eric Weisstein's World of Mathematics, Regular Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
8-regular simple graphs: this sequence (connected), A165878 (disconnected), A180260 (not necessarily connected).
Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), this sequence (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3). (End)

a(n) = A184983(n) + A181154(n).
a(n) = A180260(n) + A165878(n).
This sequence is the inverse Euler transformation of A180260.

Using the symmetry of A051031, a(15) and a(16) were appended by Jason Kimberley, Sep 25 2009

a(17)-a(22) from Andrew Howroyd, Mar 13 2020

A033301 Number of 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, 266, 1547, 10786, 88193, 805579, 8037796, 86223660, 985883873, 11946592242, 152808993767, 2056701139136, 29051369533596, 429669276147047, 6640178380127244, 107026751932268789, 1796103830404560857, 31334029441145918974, 567437704731717802783

Offset: 0

Ronald C. Read

Because the triangle A051031 is symmetric, a(n) is also the number of (n-5)-regular graphs on n vertices. - Jason Kimberley, Sep 22 2009

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

M. Meringer, Tables of Regular Graphs
M. Meringer, Erzeugung Regulärer Graphen, Diploma thesis, University of Bayreuth, January 1996. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Quartic Graph

4-regular simple graphs: A006820 (connected), A033483 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] (* Jean-François Alcover, Nov 26 2019, updated Mar 17 2020 *)

Euler transform of A006820. - Martin Fuller, Dec 04 2006

a(16) from Axel Kohnert (kohnert(AT)uni-bayreuth.de), Jul 24 2003
a(17)-a(19) from Jason Kimberley, Sep 12 2009
a(20)-a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
a(22) from Jason Kimberley, Oct 15 2011
a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 08 2020

A068933 Triangular array D(n, r) = number of disconnected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 1, 4, 2, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 12, 31, 25, 3, 1, 0, 0, 0, 0, 0

Offset: 1

David Wasserman, Mar 08 2002

A graph is called r-regular if every node has exactly r edges. Row sums give A068932.

			This sequence can be computed using the information in A068934. We'll abbreviate A068934(n, r) as C(n, r). To compute D(13, 4), note that the connected components of a 4-regular graph must have at least 5 elements. So a disconnected 13-node 4-regular graph must have two components and their sizes are either 8 and 5, or 7 and 6. So D(13, 4) = C(8, 4)*C(5, 4) + C(7, 4)*C(6, 4) = 6*1 + 2*1 = 8.
0;
1, 0;
1, 0, 0;
1, 1, 0, 0;
1, 0, 0, 0, 0;
1, 1, 1, 0, 0, 0;
1, 0, 1, 0, 0, 0, 0;
1, 1, 2, 1, 0, 0, 0, 0;
1, 0, 3, 0, 0, 0, 0, 0, 0;
1, 1, 4, 2, 1, 0, 0, 0, 0, 0;
1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0;
1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0;
1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0;
1, 1, 12, 31, 25, 3...

Jason Kimberley, Rows 1..23 of A068933 triangle, flattened.
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Cf. A051031, A068932, A068934.

D(n, r) = A051031(n, r) - A068934(n, r).

A014381 Number of connected regular graphs of degree 9 with 2n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 9, 88193, 113314233813, 281341168330848874, 1251392240942040452186674, 9854603833337765095207342173991, 134283276101750327256393048776114352985

Offset: 0

N. J. A. Sloane

Since the nontrivial 9-regular graph with the least number of vertices is K_10, there are no disconnected 9-regular graphs with less than 20 vertices. Thus for n<20 this sequence also gives the number of all 9-regular graphs on 2n vertices. - Jason Kimberley, Sep 25 2009

			The null graph on 0 vertices is vacuously connected and 9-regular; since it is acyclic, it has infinite girth. - _Jason Kimberley_, Feb 10 2011

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Regular Graph.

Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), this sequence (k=9), A014382 (k=10), A014384 (k=11).
9-regular simple graphs: this sequence (connected), A185293 (disconnected).
Connected 9-regular simple graphs with girth at least g: this sequence (g=3), A181170 (g=4).
Connected 9-regular simple graphs with girth exactly g: A184993 (g=3).

a(n) = A184993(n) + A181170(n).

a(8) appended using the symmetry of A051031 by Jason Kimberley, Sep 25 2009
a(9)-a(10) from Andrew Howroyd, Mar 13 2020
a(10) corrected and a(11)-a(12) from Andrew Howroyd, May 19 2020

A014382 Number of connected regular graphs of degree 10 with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 10, 540, 805579, 2585136741, 9799685588961, 42700033549946255, 214755319657939505396, 1251392240942040452186675, 8462215143144463851848329660, 66398444413512642732641312352087, 603696608755863722277922645973602843, 6346188247029220928621633703157327186101

Offset: 0

N. J. A. Sloane

Since the nontrivial 10-regular graph with the least number of vertices is K_11, there are no disconnected 10-regular graphs with less than 22 vertices. Thus for n<22 this sequence also gives the number of all 10-regular graphs on n vertices. - Jason Kimberley, Sep 25 2009

			The null graph on 0 vertices is vacuously connected and 10-regular; since it is acyclic, it has infinite girth. - _Jason Kimberley_, Feb 10 2011

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast Generation of Regular Graphs and Construction of Cages, Journal of Graph Theory, 30 (1999), 137-146.
Eric Weisstein's World of Mathematics, Regular Graph.

10-regular simple graphs: this sequence (connected), A185203 (disconnected).

Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), this sequence (k=10), A014384 (k=11).

Using the symmetry of A051031, a(16) and a(17) from Jason Kimberley, Sep 25 2009 and Jan 03 2011
a(18)-a(21) from Andrew Howroyd, Mar 13 2020
a(22)-a(24) from Andrew Howroyd, May 19 2020

cp's OEIS Frontend

A008483 Number of partitions of n into parts >= 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A005176 Number of regular graphs with n unlabeled nodes.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A068934 Triangular array C(n, r) = number of connected r-regular graphs with n nodes, 0 <= r < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A014378 Number of connected regular graphs of degree 8 with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A033301 Number of 4-valent (or quartic) graphs with n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A068933 Triangular array D(n, r) = number of disconnected r-regular graphs with n nodes, 0 <= r < n.