A057500 -id:A057500 - cp's OEIS frontend

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

1, 0, 1, 0, 2, 1, 1, 9, 6, 1, 15, 68, 48, 12, 1, 222, 720, 510, 150, 20, 1, 3670, 9738, 6825, 2180, 360, 30, 1, 68820, 159628, 110334, 36960, 6895, 735, 42, 1, 1456875, 3067320, 2090760, 721560, 145530, 17976, 1344, 56, 1, 34506640, 67512798, 45422928, 15989232, 3402756, 463680, 40908, 2268, 72, 1

Offset: 0

Gus Wiseman, Jan 10 2024

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Triangle begins:
      1
      0      1
      0      2      1
      1      9      6      1
     15     68     48     12      1
    222    720    510    150     20      1
   3670   9738   6825   2180    360     30      1
  68820 159628 110334  36960   6895    735     42      1
Row n = 3 counts the following loop-graphs:
  {{1,2},{1,3},{2,3}}  {{1},{1,2},{1,3}}  {{1},{2},{1,3}}  {{1},{2},{3}}
                       {{1},{1,2},{2,3}}  {{1},{2},{2,3}}
                       {{1},{1,3},{2,3}}  {{1},{3},{1,2}}
                       {{2},{1,2},{1,3}}  {{1},{3},{2,3}}
                       {{2},{1,2},{2,3}}  {{2},{3},{1,2}}
                       {{2},{1,3},{2,3}}  {{2},{3},{1,3}}
                       {{3},{1,2},{1,3}}
                       {{3},{1,2},{2,3}}
                       {{3},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Axiom of choice.

Column k = n-1 is A002378.
The case of a unique choice is A061356, row sums A000272.
Column k = 0 is A137916, unlabeled version A137917.
Row sums appear to be A333331.
The complement has row sums A368596, covering case A368730.
The unlabeled version is A368926.
Without the choice condition we have A368928, A116508, A367863, A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]],{n,0,5},{k,0,n}]

T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024

E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 1335, 47460, 1651230, 59636640, 2284113762, 93498908580, 4099070635935, 192365988161490, 9646654985111430, 515736895712230192, 29321225548502776980, 1768139644819077541440, 112805126206185257070660, 7595507651522103787077270, 538504704005397535690160274

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(5) = 30 counts all permutations of the graph {{1,2},{1,3},{1,4},{2,3},{2,4}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version without the choice condition is A116508, covering A367863.
The complement is A137916.
Allowing any number of edges gives A367867, covering A367868.
The version with loops is A368596, covering A368730, unlabeled A368835.
For set-systems we have A368600, for any number of edges A367903.
The covering case is A369144.
A006125 counts simple graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000085, A057500, A062740, A129271, A133686, A333331, A367769, A367869, A367901, A367907.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A116508(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

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.

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.
Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]

Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A369144 Number of labeled simple graphs with n edges covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 90, 4935, 200970, 7636860, 291089610, 11459170800, 471932476290, 20447369179380, 933942958593645, 44981469288560805, 2282792616992648670, 121924195590795244920, 6843305987751060036720, 403003907531795513467260, 24861219342100679072572470

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(6) = 90 counts all permutations of the (non-connected) graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The covering complement is counted by A137916.
Without the choice condition we have A367863, covering case of A116508.
Allowing any number of edges gives A367868, covering case of A367867.
With loops we have A368730, covering case of A368596, unlabeled A368835.
This is the covering case of A369143.
A003465 counts covering set-systems, unlabeled A055621.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems, unlabeled A000612.
A322661 counts covering loop-graphs, connected A062740.
Cf. A000085, A057500, A129271, A133686, A333331, A367869, A367901, A367903, A367907, A368600.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]==0&]],{n,0,6}]

a(n) = A367863(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 4905, 110715, 2864148, 83838720, 2744568522, 99463408335, 3955626143040, 171344363805582, 8031863998136355, 405150528051451000, 21884686370917378050, 1260420510502767861840, 77105349570138633021624, 4993117552678619556356085

Offset: 0

Gus Wiseman, Feb 17 2024

nonn

			The a(0) = 0 through a(4) = 15 graphs:
  {}  .  .  {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{1,4},{2,3}}
                                 {{1,2},{1,3},{1,4},{2,4}}
                                 {{1,2},{1,3},{1,4},{3,4}}
                                 {{1,2},{1,3},{2,3},{2,4}}
                                 {{1,2},{1,3},{2,3},{3,4}}
                                 {{1,2},{1,3},{2,4},{3,4}}
                                 {{1,2},{1,4},{2,3},{2,4}}
                                 {{1,2},{1,4},{2,3},{3,4}}
                                 {{1,2},{1,4},{2,4},{3,4}}
                                 {{1,2},{2,3},{2,4},{3,4}}
                                 {{1,3},{1,4},{2,3},{2,4}}
                                 {{1,3},{1,4},{2,3},{3,4}}
                                 {{1,3},{1,4},{2,4},{3,4}}
                                 {{1,3},{2,3},{2,4},{3,4}}
                                 {{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500.
This is the connected case of A116508.
Allowing any number of edges gives A287689.
Counting only covered vertices gives A370318.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187.
A369192 counts graphs with at most n edges, covering A369191.
Cf. A000169, A001862, A054780, A062740, A129271, A367863, A369193, A369197.

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

a(n)=n!*polcoef(polcoef(exp(x + O(x*x^n))*(1 + log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k,2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

a(n) = n!*[x^n][y^n] exp(x)*(1 + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

a(8) onwards from Andrew Howroyd, Feb 19 2024

A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 5, 27, 216, 2311, 30988, 499919, 9431026, 203743252, 4960335470, 134382267082, 4009794148101, 130668970606412, 4617468180528235, 175867725701333896, 7182126650899080024, 313063334893103361130, 14507460736615554141354, 712192629608088061633746

Offset: 0

Gus Wiseman, Nov 28 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A322147. The unlabeled version is A191970.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A291842 (planar case), A321254, A322114, A322115.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[multsubs[Range[n+1],2],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 18, 6, 0, 0, 192, 72, 48, 0, 0, 2500, 960, 720, 540, 0, 0, 38880, 15000, 11520, 9720, 7680, 0, 0, 705894, 272160, 210000, 181440, 161280, 131250, 0, 0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0

Offset: 0

Peter Luschny, Jan 11 2024

A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.

			Triangle starts:
  [0] [0]
  [1] [0,        0]
  [2] [0,        2,       0]
  [3] [0,       18,       6,       0]
  [4] [0,      192,      72,      48,      0]
  [5] [0,     2500,     960,     720,     540,       0]
  [6] [0,    38880,   15000,   11520,    9720,    7680,       0]
  [7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
  [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]

John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.

T(n, 1) = A066274(n) for n >= 1.
T(n, 1)/(n - 1) = A000169(n) for n >= 2.
T(n, n - 1) = 2*A081133(n) for n >= 1.
Sum_{k=0..n} T(n, k) = A001864(n).
(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).

A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)

def T(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])

cp's OEIS Frontend

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

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A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

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A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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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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A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

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Mathematica

PARI

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A368186 Number of n-covers of an unlabeled n-set.

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Mathematica

PARI

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A369144 Number of labeled simple graphs with n edges covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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Mathematica

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A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

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A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).