A006129 -id:A006129 - cp's OEIS frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

1, 2, 5, 12, 30, 73, 185, 467, 1207, 3147, 8329, 22245, 60071, 163462, 448277, 1236913, 3432327, 9569352, 26792706, 75288346, 212249873, 600069431, 1700826842, 4831722294, 13754016792, 39224295915, 112048279650, 320563736148, 918388655873, 2634460759783, 7566000947867

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024

			The a(0) = 1 through a(3) = 12 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{1},{2}}    {{1},{2}}
             {{1},{1,2}}  {{1},{1,2}}
                          {{1},{2,3}}
                          {{1,2},{1,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Without the choice condition we get A000666, labeled A006125 (shifted left).
The case of a unique choice is A087803, labeled A088957.
Without loops we have A134964, labeled A133686 (covering A367869).
For exactly n edges and no loops we have A137917, labeled A137916.
The labeled version is A368927, covering A369140.
The labeled complement is A369141, covering A369142.
For exactly n edges we have A368984, labeled A333331 (maybe).
The complement for exactly n edges is A368835, labeled A368596.
The complement is counted by A369146, labeled A369141 (covering A369142).
The covering case is A369200.
The complement for exactly n edges and no loops is A369201, labeled A369143.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable labeled graphs, covering A367868.
A368927 counts choosable labeled loop-graphs, covering A369140.
Cf. A000088, A006649, A062740, A140638, A368924, A369194.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Partial sums of A369200.

Euler transform of A369289. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A372175 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2.

Original entry on oeis.org

1, 0, 1, 3, 1, 19, 15, 0, 6, 0, 0, 0, 1, 155, 232, 15, 190, 0, 0, 70, 50, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 24 2024

A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.

			Triangle begins (zeros shown as dots):
  1
  .
  1
  3 1
  19 15 . 6 ... 1
  155 232 15 190 .. 70 50 .... 30 15 .......... 10 .............. 1
Row n = 4 counts the following graphs:
  12,34     12,13,14,23  .  12,13,14,23,24  .  .  .  12,13,14,23,24,34
  13,24     12,13,14,24     12,13,14,23,34
  14,23     12,13,14,34     12,13,14,24,34
  12,13,14  12,13,23,24     12,13,23,24,34
  12,13,24  12,13,23,34     12,14,23,24,34
  12,13,34  12,13,24,34     13,14,23,24,34
  12,14,23  12,14,23,24
  12,14,34  12,14,23,34
  12,23,24  12,14,24,34
  12,23,34  12,23,24,34
  12,24,34  13,14,23,24
  13,14,23  13,14,23,34
  13,14,24  13,14,24,34
  13,23,24  13,23,24,34
  13,23,34  14,23,24,34
  13,24,34
  14,23,24
  14,23,34
  14,24,34

Row lengths are A002807 + 1.
Row sums are A006129, unlabeled A002494.
Column k = 0 is A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372167 (non-covering A372170), unlabeled A372173 (non-covering A263340).
The non-covering version is A372176.
Column k = 1 is A372195 (non-covering A372193, for triangles A372171), unlabeled A372191 (non-covering A236570, for triangles A372174).
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A121251, A137916, A137917, A372172, A372194.

cycles[g_]:=Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@g,{k}],Min@@#==First[#]&&And@@Table[MemberQ[Sort/@g,Sort[{#[[i]], #[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[g]}];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[cycles[#]]==2k&]], {n,0,5},{k,0,Length[cycles[Subsets[Range[n],{2}]]]/2}]

A372194 Number of unlabeled graphs with n vertices and a unique triangle.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 23, 102, 526, 3624, 32240, 382095, 5986945

Offset: 0

Gus Wiseman, Apr 24 2024

The labeled version is A372172.

			Representatives of the a(3) = 1 through a(6) = 23 graphs:
    12,13,23    12,13,23       12,13,23             12,13,23
                14,23,24,34    12,34,35,45          12,34,35,45
                               14,23,24,34          14,23,24,34
                               12,25,34,35,45       12,25,34,35,45
                               14,25,34,35,45       12,36,45,46,56
                               15,25,34,35,45       13,23,45,46,56
                               12,14,25,34,35,45    14,25,34,35,45
                                                    15,25,34,35,45
                                                    12,14,25,34,35,45
                                                    12,23,36,45,46,56
                                                    13,23,36,45,46,56
                                                    13,25,36,45,46,56
                                                    13,26,36,45,46,56
                                                    14,25,36,45,46,56
                                                    15,26,36,45,46,56
                                                    16,26,36,45,46,56
                                                    12,13,25,36,45,46,56
                                                    12,13,26,36,45,46,56
                                                    13,23,25,36,45,46,56
                                                    14,23,25,36,45,46,56
                                                    16,23,25,36,45,46,56
                                                    13,14,23,25,36,45,46,56
                                                    13,15,23,25,36,45,46,56

Gus Wiseman, The a(3) = 1 through a(6) = 23 graphs with a unique triangle.

For no triangles we have A006785, covering A372169.
Column k = 1 of A263340, covering A372173.
The labeled version is A372172.
The covering case is A372174, labeled A372171.
For all cycles (not just triangles): A236570, A372193, A372191, A372195.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002494 counts unlabeled covering graphs, labeled A006129.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A137917, A137918, A140637, A144958, A213434, A372168, A372170.

geng $n | countg -T1 # Georg Grasegger, Aug 03 2024

First differences are A372174.

a(11)-a(12) added by Georg Grasegger, Aug 03 2024

A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A000701 at a(11) = 28, A000701(11) = 27
A vertical section is a partial Young diagram with at most one square in each row.
Conjecture: a(n) is the number of non-half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(2) = 1 through a(9) = 14 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the non-half-loop-graphical partitions up to n = 9:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (331)  (71)    (81)
                               (421)  (422)   (432)
                               (511)  (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (4311)
                                              (5211)
                                              (6111)
For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
  {{1,1},{1,2},{1,3},{2,2}}
  {{1},{2},{1,1},{1,2},{2,3}}
  {{1},{2},{1,1},{1,3},{2,2}}
  {{1},{3},{1,1},{1,2},{2,2}}
None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

The complement is counted by A321729.
Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339655.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 counts graphical partitions of 2n into k parts.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is zero, where m is monomial and e is elementary symmetric functions.

a(n) = A000041(n) - A321729(n).

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=Length[Union@@#]&]],{n,0,5}]

Binomial transform of A369194.

A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 3, 7, 19, 48, 135, 373, 1085, 3184, 9590, 29258, 90833, 285352, 908006, 2919953, 9487330, 31111997, 102934602, 343389708, 1154684849, 3912345408, 13353796977, 45906197103, 158915480378, 553897148543, 1943627750652, 6865605601382, 24411508473314, 87364180212671, 314682145679491

Offset: 0

Gus Wiseman, Feb 16 2024

nonn

			The a(0) = 1 through a(4) = 19 loop-graph edge sets (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}          {{1,2},{3,4}}
             {{1},{2}}    {{1,2},{1,3}}        {{1},{2},{3,4}}
             {{1},{1,2}}  {{1},{2},{3}}        {{1},{1,2},{3,4}}
                          {{1},{2},{1,3}}      {{1},{2,3},{2,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{3},{4}}
                          {{1},{1,2},{2,3}}    {{1,2},{1,3},{1,4}}
                          {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{2,4}}
                                               {{1},{2},{3},{1,4}}
                                               {{1},{2},{1,2},{3,4}}
                                               {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{2},{1,3},{3,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{1},{2,3},{2,4},{3,4}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

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

The case of equality is A368599, covering case of A368598.
The labeled version is A369194, covering case of A066383.
This is the covering case of A370168.
The loopless version is the covering case of A370315, labeled A369192.
This is the loopless version is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000666, A005703, A014068, A062740, A116508, A368597, A368927, A369141, A369196, A369197, A369199.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&Length[#]<=n&]]],{n,0,5}]

\\ G defined in A070166.
a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024

a(7) onwards from Andrew Howroyd, Feb 19 2024

A079491 Numerator of Sum_{k=0..n} binomial(n,k)/2^(k*(k-1)/2).

Original entry on oeis.org

1, 2, 7, 45, 545, 12625, 564929, 49162689, 8361575425, 2789624383745, 1830776926245889, 2368773751202917377, 6053217182280501452801, 30595465072175429929979905, 306239118989330960523869667329, 6076268165073202122463201684865025

Offset: 0

N. J. A. Sloane, Jan 20 2003

Conjecture: Also the number of loop-graphs on n vertices without any non-loop edge having loops at both ends, with formula a(n) = Sum_{k=0..n} binomial(n,k) 2^(k*(n-k) + binomial(k,2)). The unlabeled version is A339832. - Gus Wiseman, Jan 25 2024

The above conjecture is true since (n-k)*k + binomial(n-k,2) = binomial(n,2) - binomial(k,2) and A006125 gives the denominators for this sequence. - Andrew Howroyd, Feb 20 2024

			1, 2, 7/2, 45/8, 545/64, 12625/1024, 564929/32768, 49162689/2097152, ...

D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, CRC Press, 1999, p. 113.

G. C. Greubel, Table of n, a(n) for n = 0..80

Denominators are in A006125.
Cf. A079492.
The unlabeled version is A339832 (loop-graphs interpretation).
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts labeled loop-graphs, covering A322661.
A006129 counts labeled covering graphs, connected A001187.

[Numerator( (&+[Binomial(n,k)/2^Binomial(k,2): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, Jun 19 2019

f := n->add(binomial(n,k)/2^(k*(k-1)/2),k=0..n);

Table[Numerator[Sum[Binomial[n,k]/2^Binomial[k,2], {k,0,n}]], {n,0,20}] (* G. C. Greubel, Jun 19 2019 *)

{a(n)=n!*polcoeff(sum(k=0, n, exp(2^k*x +x*O(x^n))*2^(k*(k-1)/2)*x^k/k!), n)} \\ Paul D. Hanna, Sep 14 2009

a(n) = sum(k=0, n, binomial(n,k)*2^(binomial(n,2)-binomial(k,2))) \\ Andrew Howroyd, Feb 20 2024

[numerator( sum(binomial(n,k)/2^binomial(k,2) for k in (0..n)) ) for n in (0..20)] # G. C. Greubel, Jun 19 2019

E.g.f.: Sum_{n>=0} a(n)*x^n/n! = Sum_{n>=0} exp(2^n*x)*2^(n(n-1)/2)*x^n/n!. - Paul D. Hanna, Sep 14 2009

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(binomial(n,2)-binomial(k,2)). - Andrew Howroyd, Feb 20 2024

A326329 Number of simple graphs covering {1..n} with no crossing or nesting edges.

Original entry on oeis.org

1, 0, 1, 4, 13, 44, 149, 504, 1705, 5768, 19513, 66012

Offset: 0

Gus Wiseman, Jun 27 2019

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Is this (apart from offsets) the same as A073717? - R. J. Mathar, Jul 04 2019

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
Gus Wiseman, The a(5) = 44 covering simple graphs with no crossing or nesting edges.

The case for set partitions is A001519.
Covering simple graphs are A006129.
The case with just nesting or just crossing edges forbidden is A324169.
The binomial transform is the non-covering case A326244.
Cf. A000108, A006125, A007297, A054726, A099947, A324327, A326279, A326293.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

A327129 Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 1, 2, 35, 2804

Offset: 0

Gus Wiseman, Aug 27 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			The a(3) = 35 set-systems:
  {123}  {1}{12}{23}   {1}{2}{12}{13}   {1}{2}{3}{12}{13}
         {1}{13}{23}   {1}{2}{12}{23}   {1}{2}{3}{12}{23}
         {1}{2}{123}   {1}{2}{13}{23}   {1}{2}{3}{13}{23}
         {1}{3}{123}   {1}{2}{3}{123}   {1}{2}{3}{12}{123}
         {2}{12}{13}   {1}{3}{12}{13}   {1}{2}{3}{13}{123}
         {2}{13}{23}   {1}{3}{12}{23}   {1}{2}{3}{23}{123}
         {2}{3}{123}   {1}{3}{13}{23}
         {3}{12}{13}   {2}{3}{12}{13}
         {3}{12}{23}   {2}{3}{12}{23}
         {1}{23}{123}  {2}{3}{13}{23}
         {2}{13}{123}  {1}{2}{13}{123}
         {3}{12}{123}  {1}{2}{23}{123}
                       {1}{3}{12}{123}
                       {1}{3}{23}{123}
                       {2}{3}{12}{123}
                       {2}{3}{13}{123}

The restriction to simple graphs is A327079, with non-covering version A327231.
The version for spanning edge-connectivity is A327145, with BII-numbers A327111.
The BII-numbers of these set-systems are A327099.
The non-covering version is A327196.
Cf. A003465, A006129, A263296, A322395, A323818, A327071, A327149.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,3}]

Inverse binomial transform of A327196.

A339844 Number of distinct sorted degree sequences among all n-vertex loop-graphs.

Original entry on oeis.org

1, 2, 6, 16, 51, 162, 554, 1918, 6843, 24688, 90342, 333308, 1239725

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(0) = 1 through a(3) = 16 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)
      (2)  (0,2)  (0,0,2)
           (1,1)  (0,1,1)
           (1,3)  (0,1,3)
           (2,2)  (0,2,2)
           (3,3)  (0,3,3)
                  (1,1,2)
                  (1,1,4)
                  (1,2,3)
                  (1,3,4)
                  (2,2,2)
                  (2,2,4)
                  (2,3,3)
                  (2,4,4)
                  (3,3,4)
                  (4,4,4)
For example, the loop-graphs
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{3,3},{1,3}}
  {{1,1},{2,2},{3,3},{2,3}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{3,3},{1,2},{2,3}}
  {{2,2},{3,3},{1,2},{1,3}}
all have degrees y = (3,3,2), so y is counted under a(3).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
The version without loops is A004251, with covering case A095268.
The half-loop version is A029889, with covering case A339843.
Loop-graphs are counted by A322661 and ranked by A320461 and A340020.
The covering case (no zeros) is A339845.
A007717 counts unlabeled multiset partitions into pairs.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A101048 counts partitions into semiprimes.
A339655 counts non-loop-graphical partitions of 2n.
A339656 counts loop-graphical partitions of 2n.
A339659 counts graphical partitions of 2n into k parts.
Cf. A001358, A006125, A006129, A062740, A338898, A339841.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{1,2}]/.{x_Integer}:>{x,x}]]],{n,0,5}]

a(7)-a(12) from Andrew Howroyd, Jan 10 2024

cp's OEIS Frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

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A372175 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2.

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A372194 Number of unlabeled graphs with n vertices and a unique triangle.

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nauty

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A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

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A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

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PARI

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A079491 Numerator of Sum_{k=0..n} binomial(n,k)/2^(k*(k-1)/2).

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