A054921 -id:A054921 - cp's OEIS frontend

A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

1, 1, 1, 2, 5, 18, 98, 779, 10589, 255790, 11633297, 1004417286, 163944008107, 50324877640599, 29001521193534445, 31396727025729968365, 63969154112074956299242, 245871360738448777028919520, 1787330701747389106609369225312, 24636017249593067184544456944967278

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint
Gus Wiseman, The a(5) = 18 simple connected graphs whose bridges are all leaves.

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322394, A322395.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
bridgelessGraphs(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); sSolve( gc + gcr^2/2 - sRaise(gcr,2)/2, x*sv(1)*sExp(gcr) )}
cycleIndexSeries(n)={1+sSubstOp(bridgelessGraphs(n), symGroupSeries(n))}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 31 2020

a(6)-a(10) from Andrew Howroyd, Dec 08 2018

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2020

A283755 Irregular triangular array read by rows: T(n,k) = number of non-isomorphic unlabeled connected graphs with loops on n nodes and with k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 2, 6, 11, 13, 10, 5, 2, 1, 3, 14, 35, 61, 75, 68, 49, 28, 13, 5, 2, 1, 6, 33, 112, 262, 463, 625, 684, 620, 468, 301, 170, 82, 35, 14, 5, 2, 1, 11, 81, 347, 1059, 2458, 4565, 7018, 9122, 10186, 9878, 8366, 6219, 4085, 2377, 1232, 582, 251, 98, 37, 14, 5, 2, 1, 23, 204, 1085, 4091, 12014, 28779, 58162, 101315, 154484, 208321, 250120, 268649, 258994

Offset: 1

Marko Riedel, Mar 15 2017

The range for the subindex k is from n-1 to n(n+1)/2.

			First rows are:
1,  1;
1,  1,  1;
1,  3,  3,  2,  1;
2,  6, 11, 13, 10,  5,  2,  1;
3, 14, 35, 61, 75, 68, 49, 28, 13, 5, 2, 1;

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko R. Riedel, Number of distinct connected digraphs
Marko Riedel, Maple code for A070166 and A283755.

Row sums are A054921.

Cf. A000666, A070166, A322114.

T(n,k) = A322114(k,n). - Andrew Howroyd, Oct 23 2019

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128

Offset: 0

Gus Wiseman, May 24 2018

nonn

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

A322659 Number of connected regular simple graphs on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 4, 13, 146, 826, 44808, 1074557, 155741296, 10381741786, 6939251270348, 2203360264480750, 4186526735251514044, 3747344007864300197810, 35041787059621536192399824, 156277111373298355107598128061, 4142122641757597729416733678931968

Offset: 1

Gus Wiseman, Dec 22 2018

nonn

A graph is regular if all vertices have the same degree.

Gus Wiseman, The a(5) = 13 connected regular simple graphs.
Gus Wiseman, The a(6) = 146 connected regular simple graphs.

Row sums of A324163.

Cf. A054921, A059441, A062740, A295193, A299353, A305078, A319190, A319612, A319729, A322635.

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]]]]]]]]];
Table[If[n==1,1,Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],SameQ@@Length/@Split[Sort[Join@@#]],Length[csm[#]]==1]&]]],{n,6}]

a(8)-a(15) from Andrew Howroyd, Dec 23 2018

a(16)-a(18) from Andrew Howroyd, Sep 02 2019

A318590 Number of connected balanced simple signed graphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 5, 28, 177, 1982, 33997, 1020516, 54570672, 5347070228, 967135763525, 324200029119318, 202046821340636691, 234878262433630160622, 511060736355598412146405, 2088401066728281847415734793, 16079824271822965645002329491423, 233994776259866281916838227225733732

Offset: 0

Andrew Howroyd, Sep 25 2018

nonn

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

Cf. A005142, A034892, A053465, A054919, A054921.

a(2*n+1) = A054921(2*n+1)/2, a(2*n) = (a(n) + A054919(n) + A054921(2*n) - A054921(n))/2.

Inverse Euler transform of A034892.

A322367 Number of disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303

Offset: 0

Gus Wiseman, Dec 04 2018

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The a(3) = 2 through a(9) = 27 disconnected integer partitions:
  (21)   (31)    (32)     (51)      (43)       (53)        (54)
  (111)  (211)   (41)     (321)     (52)       (71)        (72)
         (1111)  (221)    (411)     (61)       (332)       (81)
                 (311)    (2211)    (322)      (431)       (432)
                 (2111)   (3111)    (331)      (521)       (441)
                 (11111)  (21111)   (421)      (611)       (522)
                          (111111)  (511)      (3221)      (531)
                                    (2221)     (3311)      (621)
                                    (3211)     (4211)      (711)
                                    (4111)     (5111)      (3222)
                                    (22111)    (22211)     (3321)
                                    (31111)    (32111)     (4221)
                                    (211111)   (41111)     (4311)
                                    (1111111)  (221111)    (5211)
                                               (311111)    (6111)
                                               (2111111)   (22221)
                                               (11111111)  (32211)
                                                           (33111)
                                                           (42111)
                                                           (51111)
                                                           (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]!=1&]],{n,20}]

A322393 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0, 6, 1, 0, 0, 0, 0, 7, 1, 2, 1, 0, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0, 17, 1, 2, 1, 1, 0, 0, 0, 0, 27, 1, 1, 1, 0, 0, 0, 0, 0, 0, 34, 1, 3, 2, 1, 1, 0, 0, 0, 0, 0, 54, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 1, 4, 4, 3, 1, 1

Offset: 0

Gus Wiseman, Dec 06 2018

The edge connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			Triangle begins:
   1
   0  1
   1  1  0
   2  1  0  0
   3  1  1  0  0
   6  1  0  0  0  0
   7  1  2  1  0  0  0
  14  1  0  0  0  0  0  0
  17  1  2  1  1  0  0  0  0
  27  1  1  1  0  0  0  0  0  0
  34  1  3  2  1  1  0  0  0  0  0
  54  2  0  0  0  0  0  0  0  0  0  0
  63  1  4  4  3  1  1  0  0  0  0  0  0
Row 6 {7, 1, 2, 1} counts the following integer partitions:
  (51)      (6)  (33)  (222)
  (321)          (42)
  (411)
  (2211)
  (3111)
  (21111)
  (111111)

Wikipedia, k-edge-connected graph

Row sums are A000041. First column is A322367. Second column is A322391.

Cf. A013922, A054921, A095983, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322387.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
edgeConn[y_]:=Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]
Table[Length[Select[IntegerPartitions[n],edgeConn[#]==k&]],{n,10},{k,0,n}]

A322368 Heinz numbers of disconnected integer partitions.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

Differs from A289509 in having 1 and lacking 2, 195, 455, 555, 585...
Also positions of entries > 1 in A305079.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The sequence of all disconnected integer partitions begins: (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).

Cf. A054921, A218970, A286518, A290103, A304714, A304716, A305078, A305079, A322306, A322307, A322338, A322367, A322369.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[200],Length[csm[primeMS/@primeMS[#]]]>1&]

A322369 Number of strict disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 10, 16, 17, 22, 26, 33, 36, 48, 52, 64, 76, 90, 101, 125, 142, 166, 192, 225, 250, 302, 339, 393, 451, 515, 581, 675, 762, 866, 985, 1122, 1255, 1441, 1612, 1823, 2059, 2318, 2591, 2930, 3275, 3668, 4118, 4605, 5125, 5749

Offset: 0

Gus Wiseman, Dec 04 2018

nonn

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The a(3) = 1 through a(11) = 10 strict disconnected integer partitions:
  (2,1)  (3,1)  (3,2)  (5,1)    (4,3)    (5,3)    (5,4)    (7,3)      (6,5)
                (4,1)  (3,2,1)  (5,2)    (7,1)    (7,2)    (9,1)      (7,4)
                                (6,1)    (4,3,1)  (8,1)    (5,3,2)    (8,3)
                                (4,2,1)  (5,2,1)  (4,3,2)  (5,4,1)    (9,2)
                                                  (5,3,1)  (6,3,1)    (10,1)
                                                  (6,2,1)  (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,4,1)
                                                                      (7,3,1)
                                                                      (8,2,1)
                                                                      (5,3,2,1)

Cf. A054921, A218970, A286518, A304714, A304716, A305078, A305079, A322306, A322307, A322335, A322337, A322338, A322367, A322368.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,Length[zsm[#]]!=1]&]],{n,30}]

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

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

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

cp's OEIS Frontend

A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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A283755 Irregular triangular array read by rows: T(n,k) = number of non-isomorphic unlabeled connected graphs with loops on n nodes and with k edges.

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A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

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A322659 Number of connected regular simple graphs on n labeled vertices.

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A318590 Number of connected balanced simple signed graphs on n unlabeled nodes.

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A322367 Number of disconnected or empty integer partitions of n.

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A322393 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.

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A322368 Heinz numbers of disconnected integer partitions.

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A322369 Number of strict disconnected or empty integer partitions of n.

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