A001187 -id:A001187 - cp's OEIS frontend

A054941 Number of weakly connected oriented graphs on n labeled nodes.

1, 2, 20, 624, 55248, 13982208, 10358360640, 22792648882176, 149888345786341632, 2952810709943411146752, 174416705255313941476193280, 30901060796613886817249881227264, 16422801513633911416125344647746244608, 26183660776604240464418800095675915958222848

Offset: 1

N. J. A. Sloane, May 24 2000

The triangle of oriented labeled graphs on n>=1 nodes with 1<=k<=n components and row sums A047656 starts:
       1;
       2,      1;
      20,      6,     1;
     624,     92,    12,   1;
   55248,   3520,   260,  20,  1;
13982208, 354208, 11880, 580, 30, 1; - R. J. Mathar, Apr 29 2019

Seiichi Manyama, Table of n, a(n) for n = 1..65
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Row sums of A350732.
The unlabeled version is A086345.
Cf. A001187 (graphs), A003027 (digraphs), A350730 (strongly connected).

m:=30;
f:= func< x | (&+[3^Binomial(n,2)*x^n/Factorial(n) : n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( Log(f(x)) ))); // G. C. Greubel, Apr 28 2023

nn=20; s=Sum[3^Binomial[n,2]x^n/n!,{n,0,nn}];
Drop[Range[0,nn]! CoefficientList[Series[Log[s]+1,{x,0,nn}],x],1] (* Geoffrey Critzer, Oct 22 2012 *)

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, 3^binomial(k, 2)*x^k/k!)))) \\ Seiichi Manyama, May 18 2019

m=30
def f(x): return sum(3^binomial(n,2)*x^n/factorial(n) for n in range(m+4))
def A054941_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( log(f(x)) ).egf_to_ogf().list()
a=A054941_list(40); a[1:] # G. C. Greubel, Apr 28 2023

E.g.f.: log( Sum_{n >= 0} 3^binomial(n, 2)*x^n/n! ). - Vladeta Jovovic, Feb 14 2003

More terms from Vladeta Jovovic, Feb 14 2003

A095340 Total number of nodes in all labeled graphs on n nodes.

Original entry on oeis.org

1, 4, 24, 256, 5120, 196608, 14680064, 2147483648, 618475290624, 351843720888320, 396316767208603648, 885443715538058477568, 3929008913747544817795072, 34662321099990647697175478272, 608472288109550112718417538580480, 21267647932558653966460912964485513216

Offset: 1

Eric W. Weisstein, Jun 03 2004

nonn

Number of perfect matchings of an n X (n+1) Aztec rectangle with the second vertex in the topmost row removed.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7

Alois P. Heinz, Table of n, a(n) for n = 1..50
Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part I, Journal of Algebraic Combinatorics 1-2, 111-132 (1992).
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part II, Journal of Algebraic Combinatorics 1-3, 219-234 (1992).
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, Volume 6 (1999), Research Paper #R16.
Christian Krattenthaler, Schur function identities and the number of perfect matchings of Aztec holey rectangles, arXiv:math/9712204 [math.CO], 1997.
Maximilian Balthasar Mansky, Tobias Rohe, Gerhard Stenzel, Alejandro Bravo de la Serna, Santiago Londoño Castillo, Gautham Sathish, Dimitra Nikolaidou, Dmytro Bondarenko, Linus Menzel, and Claudia Linnhoff-Popien, Solving graph problems using permutation-invariant quantum machine learning, arXiv:2505.12764 [quant-ph], 2025. See p. 9.
Eric Weisstein's World of Mathematics, Graph Vertex

Cf. A006125, A103904.

[n*2^((n^2-n) div 2): n in [1..20]]; // Vincenzo Librandi, Aug 17 2015

a:= n-> n*2^(n*(n-1)/2):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 26 2013

g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; a = Drop[Range[0, 20]! CoefficientList[Series[Log[g] + 1, {x, 0, 20}], x], 1]; Table[Sum[k Binomial[n, k] 2^Binomial[n - k, 2] a[[k]], {k, 1, n}], {n, 1,20}] (* Geoffrey Critzer, Nov 12 2011 *)

PARI
```
a(n)=n*2^((n^2-n)/2)
```

a(n) = n * A006125(n).
a(n) = n * 2^(n*(n-1)/2). E.g., a(7) = 7 * 2^(7*6/2) = 7 * 2097152 = 14680064. - David Terr, Nov 08 2004
a(n) = (32*a(n-1)*a(n-3)-48*a(n-2)^2)/a(n-4). - Michael Somos, Sep 16 2005
a(n) = Sum_{k=1..n} k*C(n,k)*2^C(n-k,2)*A001187(k). The sum gives the number of rooted labeled simple graphs on n nodes. It conditions on k, 1<=k<=n, the size of the connected component that the root is in. See Harary and Palmer reference. - Geoffrey Critzer, Nov 12 2011

Edited by Ralf Stephan, Feb 21 2005

A287689 Number of (non-null) connected induced subgraphs in the n-triangular graph.

Original entry on oeis.org

1, 7, 60, 968, 31737, 2069963, 267270032, 68629753640, 35171000942697, 36024807353574279, 73784587576805254652, 302228602363365451957792, 2475873310144021668263093201, 40564787336902311168400640561083, 1329227697997490307154018925966130304

Offset: 2

Eric W. Weisstein, May 29 2017

nonn

Also the number of labeled simple graphs with n vertices whose edge-set is connected. - Gus Wiseman, Sep 11 2019

			From _Gus Wiseman_, Sep 11 2019: (Start)
The a(4) = 60 edge-sets:
  {12}  {12,13}  {12,13,14}  {12,13,14,23}  {12,13,14,23,24}
  {13}  {12,14}  {12,13,23}  {12,13,14,24}  {12,13,14,23,34}
  {14}  {12,23}  {12,13,24}  {12,13,14,34}  {12,13,14,24,34}
  {23}  {12,24}  {12,13,34}  {12,13,23,24}  {12,13,23,24,34}
  {24}  {13,14}  {12,14,23}  {12,13,23,34}  {12,14,23,24,34}
  {34}  {13,23}  {12,14,24}  {12,13,24,34}  {13,14,23,24,34}
        {13,34}  {12,14,34}  {12,14,23,24}
        {14,24}  {12,23,24}  {12,14,23,34}
        {14,34}  {12,23,34}  {12,14,24,34}
        {23,24}  {12,24,34}  {12,23,24,34}
        {23,34}  {13,14,23}  {13,14,23,24}
        {24,34}  {13,14,24}  {13,14,23,34}
                 {13,14,34}  {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}             {12,13,14,23,24,34}
                 {23,24,34}
(End)

Andrew Howroyd, Table of n, a(n) for n = 2..50
Eric Weisstein's World of Mathematics, Triangular Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph

The unlabeled version is A292300.

Cf. A001187, A006125, A327070, A327148, A327199, A327235.

Table[With[{g = GraphData[{"Triangular", n}]}, Total[Boole[ConnectedGraphQ[Subgraph[g, #]] & /@ Subsets[VertexList[g]]]]], {n, 2, 5}] - 1
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k) * (n-k-1)/2)*g[k], {k, 1, n-1}]/n]; a[n_] := Sum[Binomial[n, i]*g[i], {i, 2, n}]; Table[a[n], {n, 2, 16}] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)

seq(n)={Vec(serlaplace(exp(x + O(x*x^n))*(-x+log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Sep 11 2019

a(n) = Sum_{i=2..n} binomial(n,i) * A001187(i). - Andrew Howroyd, Jun 07 2017
E.g.f.: exp(x)*(-x + log(Sum_{k>=0} 2^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Sep 11 2019
a(n) = A006125(n) - A327199(n). - Gus Wiseman, Sep 11 2019

Terms a(9) and beyond from Andrew Howroyd, Jun 07 2017

A317632 Number of connected induced nonempty non-singleton subgraphs of labeled connected graphs with n vertices.

Original entry on oeis.org

0, 0, 1, 13, 294, 12198, 946712, 140168924, 40223263760, 22598607583376, 24999757695984960, 54630901092648916704, 236304498092496715916416, 2026201628540583716863002880, 34482826679730591694177065948928, 1166004710785628820717860509317415168

Offset: 0

Gus Wiseman, Aug 02 2018

nonn

The edges of an induced subgraph G|S are those edges of G with both ends contained in S, where S is a subset of the vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, All 294 connected induced subgraphs of labeled connected graphs with 4 vertices.

Cf. A001187, A006125, A048143, A293510, A304717, A317631, A317634, A317635.

seq(n)={
  my(p=sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n)));
  my(g=Vec(serlaplace(log(p))));
  my(q=sum(k=0, n, sum(j=2, k, binomial(k,j)*g[j]*2^(binomial(k-j, 2) + j*(k-j)))*x^k/k!, O(x*x^n)));
  Vec(serlaplace(q/p), -n-1)
} \\ Andrew Howroyd, Dec 10 2018

a(6) from Gus Wiseman, Dec 10 2018

Terms a(7) and beyond from Andrew Howroyd, Dec 10 2018

A317635 Number of connected vertex sets of clutters (connected antichains) spanning n vertices.

Original entry on oeis.org

1, 0, 1, 14, 486, 71428

Offset: 0

Gus Wiseman, Aug 02 2018

A connected vertex set in a clutter is any union of a connected subset of the edges.

			There are four connected vertex sets of {{1,2},{1,3},{2,3}}, namely {1,2,3}, {1,2}, {1,3}, {2,3}; there are three connected vertex sets of {{1,2},{1,3}}, {{1,2},{2,3}}, and {{1,3},{2,3}} each; and there is one connected vertex set of {{1,2,3}}. So we have a total of a(3) = 4 + 3 * 3 + 1 = 14 connected vertex sets.

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

Cf. A001187, A006126, A030019, A048143, A134954, A275307, A286520, A293510, A304717, A317631, A317632, A317634.

nn=5;
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]]],multijoin@@s[[c[[1]]]]]]]]];
clutQ[eds_]:=And[UnsameQ@@eds,!Apply[Or,Outer[#1=!=#2&&Complement[#1,#2]=={}&,eds,eds,1],{0,1}],Length[csm[eds]]==1];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
swell[c_]:=Union@@FixedPointList[Union[ReplaceList[#1,{_,a:{_,x_,_},_,b:{_,x_,_},_}:>Union[a,b]]]&,c]
Table[Sum[Length[swell[c]],{c,Select[stableSets[Select[Subsets[Range[n]],Length[#]>1&],Complement[#1,#2]=={}&],And[Union@@#==Range[n],clutQ[#]]&]}],{n,nn}]

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.

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

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

Original entry on oeis.org

1, 1, 2, 8, 57, 608, 8614, 151365, 3162353, 76359554, 2088663444, 63760182536, 2147325661180, 79051734050283, 3157246719905273, 135938652662043977, 6275929675565965599, 309242148569525451140, 16197470691388774460758, 898619766673014862321176, 52639402023471657682257626

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The case of equality is A367862, covering case of A116508, also A367863.
The covering case is A369191, for loop-graphs A369194.
The version counting all vertices is A369192.
The version for loop-graphs is A369196, counting all vertices A066383.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A053530, A129271, A143543, A322661, A367916.

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

Binomial transform of A369191.

A371291 Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548

Offset: 1

Gus Wiseman, Mar 27 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The empty set is not considered connected.

			The terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    2:         10 ~ {2}
    4:        100 ~ {3}
    8:       1000 ~ {4}
   10:       1010 ~ {2,4}
   16:      10000 ~ {5}
   32:     100000 ~ {6}
   34:     100010 ~ {2,6}
   36:     100100 ~ {3,6}
   38:     100110 ~ {2,3,6}
   40:     101000 ~ {4,6}
   42:     101010 ~ {2,4,6}
   44:     101100 ~ {3,4,6}
   46:     101110 ~ {2,3,4,6}
   64:    1000000 ~ {7}
  128:   10000000 ~ {8}
  130:   10000010 ~ {2,8}
  136:   10001000 ~ {4,8}
  138:   10001010 ~ {2,4,8}
  160:   10100000 ~ {6,8}
  162:   10100010 ~ {2,6,8}
  164:   10100100 ~ {3,6,8}

For prime indices of each prime index we have A305078.
The opposite version is A325118.
For binary indices of each binary index we have A326749.
The pairwise indivisible case is A371294, opposite A371445.
Positions of ones in A371452.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A000040, A000720, A001222, A305079, A326753, A371446.

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]]]]]]]]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],Length[csm[prix/@bpe[#]]]==1&]

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 84, 23, 6, 1, 6348, 470, 65, 10, 1, 7743728, 39598, 1575, 145, 15, 1, 2414572893530, 54354104, 144403, 4095, 280, 21, 1, 56130437190053299918162, 19316801997024, 218033088, 402073, 9100, 490, 28, 1

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       1
        5       3       1
       84      23       6       1
     6348     470      65      10       1
  7743728   39598    1575     145      15       1

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

First column is A048143. Row sums are A006126.
Cf. A000272, A001187, A013922, A030019, A134954, A143543, A275307, A293510.
Cf. A317634, A317635, A317671, A317672, A317677.

blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]

cp's OEIS Frontend

A054941 Number of weakly connected oriented graphs on n labeled nodes.

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A095340 Total number of nodes in all labeled graphs on n nodes.

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A287689 Number of (non-null) connected induced subgraphs in the n-triangular graph.

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A317632 Number of connected induced nonempty non-singleton subgraphs of labeled connected graphs with n vertices.

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A317635 Number of connected vertex sets of clutters (connected antichains) spanning n vertices.

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

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