A298422 -id:A298422 - cp's OEIS frontend

A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

1, 1, 1, 1, 4, 1, 1, 41, 11, 1, 1, 768, 958, 26, 1, 1, 27449, 1042642, 32596, 57, 1, 1, 1887284, 34352419335, 34359509614, 2096731, 120, 1, 1, 252522481, 72057319189324805, 1180591620442534312297, 72057594021152435, 268434467, 247, 1, 1, 66376424160

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,       1;
  1,    41,      11,     1;
  1,   768,     958,    26,  1;
  1, 27449, 1042642, 32596, 57, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..91 (rows 1..13)
Wikipedia, Hypergraph

Columns 1..4 are A000012, A006129, A302374, A302396.
Row sums are A306021.
The unlabeled version is A301922.
The connected version is A299354.
Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]

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

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

A124343 Number of rooted trees on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 21, 38, 78, 153, 314, 632, 1313, 2700, 5646, 11786, 24831, 52348, 111027, 235834, 502986, 1074739, 2303146, 4944507, 10639201, 22930493, 49511948, 107065966, 231874164, 502834328, 1091842824, 2373565195, 5165713137, 11254029616, 24542260010

Offset: 1

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

nonn

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

			The a(5) = 6 trees are ((((o)))), (o((o))), (o(oo)), ((o)(o)), (oo(o)), (oooo). - _Gus Wiseman_, Jan 25 2018

Alois P. Heinz, Table of n, a(n) for n = 1..141

Cf. A000081, A032305, A124344-A124348, A290689, A298303, A298304, A298305, A298422.

Row sums of A244657.

b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n A(n$2):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 08 2014

b[n_, i_, h_, v_] := b[n, i, h, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || nJean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 04 2014

A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 5, 50, 1713, 1101990, 68715891672, 1180735735356264714926, 170141183460507906731293351306487161569, 7237005577335553223087828975127304177495735363998991435497132228228565768846

Offset: 0

Gus Wiseman, Jun 18 2018

nonn

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

Let T be the regular triangle A299354, where column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k). Then a(n) is the sum of row n.

			The a(3) = 5 hypergraphs:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}

Wikipedia, Hypergraph

Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299354, A299471, A306020, A306021.

nn=10;Table[Sum[SeriesCoefficient[Log[Sum[x^m/m!*(-1)^(m-d)*Binomial[m,d]*2^Binomial[d,k],{m,0,n},{d,0,m}]],{x,0,n}]*n!,{k,n}],{n,nn}]

A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 6, 5, 0, 0, 0, 0, 1, 9, 13, 2, 0, 0, 0, 0, 1, 12, 28, 11, 0, 0, 0, 0, 0, 1, 16, 53, 40, 3, 0, 0, 0, 0, 0, 1, 20, 91, 109, 26, 0, 0, 0, 0, 0, 0, 1, 25, 146, 254, 116, 6, 0, 0, 0, 0, 0, 0, 1, 30, 223, 524, 387, 61, 0, 0, 0, 0, 0, 0, 0, 1, 36

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
1   0   0
1   1   0   0
1   2   0   0   0
1   4   1   0   0   0
1   6   5   0   0   0   0
1   9  13   2   0   0   0   0
1  12  28  11   0   0   0   0   0
1  16  53  40   3   0   0   0   0   0
1  20  91 109  26   0   0   0   0   0   0
1  25 146 254 116   6   0   0   0   0   0   0
1  30 223 524 387  61   0   0   0   0   0   0   0
The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).

A version with the zeroes removed is A055327.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.

irut[n_]:=irut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[irut/@c]],UnsameQ@@#&]]/@IntegerPartitions[n-1]];
Table[Length[Select[irut[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]

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

A306019 Number of non-isomorphic set-systems of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 10, 2, 17, 14, 33, 2, 167, 2, 186, 491, 785, 2, 5839, 2, 11123, 53454, 15229, 2, 1102924, 53537, 193382, 16334183, 12411062, 2, 382413555, 2, 993814248, 9763321547, 53394774, 1778595972, 402119882757, 2, 1111261718, 9674133468473, 16955983996383

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A set-system of weight n is a finite set of finite nonempty sets whose sizes sum to n.

			Non-isomorphic representatives of the a(6) = 10 set-systems:
{{1,2,3,4,5,6}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1},{2},{3},{4},{5},{6}}

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

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306020, A306021.

\\ See A331508 for T(n,k).
a(n) = {if(n==0, 1, sumdiv(n, d, if(d==1 || d==n, 1, T(n/d, d))))} \\ Andrew Howroyd, Jan 16 2024

a(p) = 2 for prime p. - Andrew Howroyd, Aug 29 2019

a(n) = Sum_{d|n} A331508(n/d, d) for n > 0. - Andrew Howroyd, Jan 16 2024

Terms a(12) and beyond from Andrew Howroyd, Sep 01 2019

A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 7, 8, 10, 2, 15, 2, 16, 11, 9, 2, 28, 7, 10, 14, 22, 2, 37, 2, 25, 18, 12, 17, 55, 2, 13, 23, 52, 2, 55, 2, 40, 51, 15, 2, 95, 13, 44, 34, 53, 2, 79, 37, 85, 41, 18, 2, 185, 2, 19, 80, 91, 54, 112, 2, 87, 56, 122, 2

Offset: 0

Gus Wiseman, Jan 19 2018

nonn

Note that n is automatically divisible by the number of parts.

			The a(9) = 4 partitions: (9), (441), (711), (111111111).

Cf. A000005, A000041, A067538, A143773, A280954, A298422, A298424, A298426.

Table[Length[Select[IntegerPartitions[n],Function[ptn,And@@(Divisible[#-1,Length[ptn]]&/@ptn)]]],{n,60}]

G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(k*i)).

A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 2, 4, 1, 0, 0, 0, 0, 3, 4, 5, 1, 0, 0, 0, 0, 2, 6, 6, 6, 1, 0, 0, 0, 0, 1, 6, 10, 9, 7, 1, 0, 0, 0, 0, 1, 5, 12, 16, 12, 8, 1, 0, 0, 0, 0, 0, 4, 13, 22, 23, 16, 9, 1, 0, 0, 0, 0, 0, 3, 14, 27, 36, 32, 20, 10, 1, 0, 0, 0, 0, 0, 2, 11

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
0   1   0
0   1   1   0
0   0   2   1   0
0   0   1   3   1   0
0   0   1   2   4   1   0
0   0   0   3   4   5   1   0
0   0   0   2   6   6   6   1   0
0   0   0   1   6  10   9   7   1   0
0   0   0   1   5  12  16  12   8   1   0
The T(9,5) = 6 transitive rooted trees: (o(o)(oo(o))), (o((oo))(oo)), (oo(o)(o(o))), (o(o)(o)(oo)), (ooo(o)((o))), (oo(o)(o)(o)).

Row sums are A290689.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A276625, A279861, A290760, A290822, A298422, A298426, A301342, A301343, A301344.

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
trat[n_]:=Select[rut[n],Complement[Union@@#,#]==={}&];
Table[Length[Select[trat[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

A306018 Number of non-isomorphic set multipartitions of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 3, 4, 9, 8, 24, 16, 51, 47, 115, 57, 420, 102, 830, 879, 2962, 298, 15527, 491, 41275, 80481, 133292, 1256, 2038182, 58671, 2386862, 24061887, 23570088, 4566, 600731285, 6843, 1303320380, 14138926716, 1182784693, 1820343112, 542834549721, 21638, 31525806080

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A set multipartition of weight n is a finite multiset of finite nonempty sets whose cardinalities sum to n.

Number of distinct binary matrices with all row sums equal and total sum n, up to row and column permutations. - Andrew Howroyd, Sep 05 2018

			Non-isomorphic representatives of the a(6) = 24 set multipartitions in which all parts have the same size:
{{1,2,3,4,5,6}}
{{1,2,3},{1,2,3}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,2},{1,2}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{3,4}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,3},{2,3}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1},{1},{1},{1},{1},{1}}
{{1},{1},{1},{2},{2},{2}}
{{1},{1},{2},{2},{2},{2}}
{{1},{1},{2},{2},{3},{3}}
{{1},{2},{2},{2},{2},{2}}
{{1},{2},{2},{3},{3},{3}}
{{1},{2},{3},{3},{3},{3}}
{{1},{2},{3},{3},{4},{4}}
{{1},{2},{3},{4},{4},{4}}
{{1},{2},{3},{4},{5},{5}}
{{1},{2},{3},{4},{5},{6}}

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

Cf. A000005, A000041, A001315, A007716, A038041, A049311, A283877, A298422, A304942, A306017, A306019, A306020, A306021, A331461.

\\ See A304942 for Blocks
a(n)={sumdiv(n, d, Blocks(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018

a(p) = A000041(p) + 1 for prime p. - Andrew Howroyd, Sep 06 2018

a(n) = Sum_{d|n} A331461(n/d, d). - Andrew Howroyd, Feb 09 2020

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

A306020 a(n) is the number of set-systems using nonempty subsets of {1,...,n} in which all sets have the same size.

Original entry on oeis.org

1, 2, 5, 16, 95, 2110, 1114237, 68723671292, 1180735735906024030715, 170141183460507917357914971986913657850, 7237005577335553223087828975127304179197147198604070555943173844710572689401

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A058673(n) <= a(n). - Lorenzo Sauras Altuzarra, Aug 10 2023

			a(3) = 16 set-systems in which all sets have the same size:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1},{2}}
  {{1},{3}}
  {{2},{3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}

Index entries for sequences related to matroids

Cf. A000005, A001315, A007716, A038041, A049311, A058673 (labeled matroids), A283877, A298422, A306017, A306018, A306019, A306021.

Row sums of A059328.

a := n -> 1-n+add(2^binomial(n, d), d = 1 .. n):
seq(a(n), n = 0 .. 10); # Lorenzo Sauras Altuzarra, Aug 11 2023

Table[1+Sum[2^Binomial[n,d]-1,{d,n}],{n,10}]

a(n) = 1 - n + sum(d = 1, n, 2^binomial(n, d)); \\ Michel Marcus, Aug 10 2023

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

cp's OEIS Frontend

A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

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Mathematica

PARI

Formula

A124343 Number of rooted trees on n nodes with thinning limbs.

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Author

Keywords

Comments

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Programs

Maple

Mathematica

Extensions

A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

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Mathematica

A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

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Programs

Maple

PARI

Formula

Extensions

A306019 Number of non-isomorphic set-systems of weight n in which all parts have the same size.

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PARI

Formula

Extensions

A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

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Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

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Author

Keywords