A301481 -id:A301481 - cp's OEIS frontend

A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024

			The a(3) = 6 set-systems in which all sets have the same size:
  {{1,2,3}}
  {{1}, {2}, {3}}
  {{1,2}, {1,3}}
  {{1,2}, {2,3}}
  {{1,3}, {2,3}}
  {{1,2}, {1,3}, {2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..14

Row sums of A299471.
The unlabeled version is A301481.
The connected version is A299353.
Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-k)*Binomial[n,k]*(1+Sum[2^Binomial[k,d]-1,{d,k}]),{k,0,n}],{n,12}]

a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)))) \\ Andrew Howroyd, Jan 16 2024

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).

Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024

A301922 Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 23, 29, 4, 1, 1, 122, 2102, 150, 5, 1, 1, 888, 7011184, 7013164, 1037, 6, 1, 1, 11302, 1788775603336, 29281354507753848, 1788782615612, 12338, 7, 1, 1, 262322, 53304526022885280592, 234431745534048893449761040648512, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Gus Wiseman, Jun 19 2018

			Triangle begins:
   1
   1   1
   1   2   1
   1   7   3   1
   1  23  29   4   1
The T(4,2) = 7 hypergraphs:
  {{1,2},{3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 1..91

Row sums are A301481. Second column is A002494.

Cf. A003465, A055621, A298422, A298426, A299471, A301481, A301920, A306017-A306021, A309858.

g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
     [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
     /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
     /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
    `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
     /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
A:= proc(n, k) A(n, k):= `if`(k>n-k, A(n, n-k), b(n$2, [], k)) end:
T:= (n, k)-> A(n, k)-A(n-1, k):
seq(seq(T(n, k), k=1..n), n=1..9);  # Alois P. Heinz, Aug 21 2019

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}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L,k); while(#L0, u=vecsort(apply(f, u)); d=lex(u,v)); !d}
Q(n,k,perm)={my(t=0); forsubset([n,k], v, t += can(Vec(v), t->perm[t])); t}
U(n,k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n,k,rep(p))); s/n!}
for(n=1, 10, for(k=1, n, print1(U(n,k)-U(n-1,k), ", ")); print) \\ Andrew Howroyd, Aug 10 2019

T(n,k) = A309858(n,k) - A309858(n-1,k). - Alois P. Heinz, Aug 21 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 09 2019

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset: 1

Gus Wiseman, Dec 17 2018

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Andrew Howroyd, Table of n, a(n) for n = 1..300

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A301920 Number of unlabeled uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 3, 10, 55, 2369, 14026242, 29284932065996223, 468863491068204425232922367146585, 1994324729204021501147398087008429476673379600542622915802043455294332

Offset: 0

Gus Wiseman, Jun 19 2018

nonn

A hypergraph is uniform if all edges have the same size.

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

Andrew Howroyd, Table of n, a(n) for n = 0..13

Row sums of A301924.

Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A301481, A306017-A306021.

Terms a(6) and beyond from Andrew Howroyd, Aug 26 2019

A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 21, 29, 4, 1, 0, 112, 2101, 150, 5, 1, 0, 853, 7011181, 7013164, 1037, 6, 1, 0, 11117, 1788775603301, 29281354507753847, 1788782615612, 12338, 7, 1, 0, 261080, 53304526022885278403, 234431745534048893449761040648508, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Gus Wiseman, Jun 19 2018

			Triangle begins:
   1
   0    1
   0    2       1
   0    6       3       1
   0   21      29       4    1
   0  112    2101     150    5 1
   0  853 7011181 7013164 1037 6 1
   ...
The T(4,2) = 6 hypergraphs:
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Row sums are A301920.
Columns k=2..3 are A001349(n > 1), A003190(n > 1).
Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
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}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
{ my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 26 2019

Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A302129 Number of unlabeled uniform connected hypergraphs of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 6, 1, 9, 10, 17, 1, 108, 1, 86, 401, 482, 1, 4469, 1, 8435, 47959, 8082, 1, 1007342, 52414, 112835, 15338453, 11899367, 1, 362657533, 1, 977129970, 9349593479, 35787684, 1771297657, 390347162497, 1, 779945988, 9360467497257, 16838238535445

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A hypergraph is uniform if all edges have the same size. The weight of a hypergraph is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(8) = 9 uniform connected hypergraphs:
  {{1,2,3,4,5,6,7,8}}
  {{1,2,3,7}, {4,5,6,7}}
  {{1,2,5,6}, {3,4,5,6}}
  {{1,3,4,5}, {2,3,4,5}}
  {{1,2}, {1,3}, {2,4}, {3,4}}
  {{1,3}, {2,4}, {3,5}, {4,5}}
  {{1,4}, {2,3}, {2,4}, {3,4}}
  {{1,4}, {2,5}, {3,5}, {4,5}}
  {{1,5}, {2,5}, {3,5}, {4,5}}

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

Cf. A000005, A007716, A038041, A038041, A048143, A299353, A301481, A301920, A301924, A306019, A306021, A331508.

\\ See A331508 for T(n, k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
a(n) = {if(n==0, 1, sumdiv(n, d, if(d==1 || d==n, d==1, InvEulerT(vector(d, i, T(n/d, i)))[d] )))} \\ Andrew Howroyd, Jan 16 2024

a(p) = 1 for prime p. - Andrew Howroyd, Jan 16 2024

a(11) onwards from Andrew Howroyd, Jan 16 2024

cp's OEIS Frontend

A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

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A301922 Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices.

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A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

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A301920 Number of unlabeled uniform connected hypergraphs spanning n vertices.

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A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

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PARI

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A302129 Number of unlabeled uniform connected hypergraphs of weight n.

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