A306017 -id:A306017 - cp's OEIS frontend

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

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

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).

A318951 Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 14, 5, 1, 1, 9, 44, 53, 7, 1, 1, 12, 129, 458, 198, 11, 1, 1, 16, 316, 3411, 5929, 782, 15, 1, 1, 20, 714, 19865, 145168, 96073, 3111, 22, 1, 1, 25, 1452, 95214, 2459994, 9283247, 1863594, 12789, 30, 1, 1, 30, 2775, 383714, 30170387, 537001197, 833593500, 42430061, 53836, 42, 1

Offset: 1

Andrew Howroyd, Sep 05 2018

			Array begins:
================================================================
n\k| 0  1    2       3         4            5              6
---|------------------------------------------------------------
1  | 1  1    1       1         1            1              1 ...
2  | 1  2    4       6         9           12             16 ...
3  | 1  3   14      44       129          316            714 ...
4  | 1  5   53     458      3411        19865          95214 ...
5  | 1  7  198    5929    145168      2459994       30170387 ...
6  | 1 11  782   96073   9283247    537001197    19578605324 ...
7  | 1 15 3111 1863594 833593500 189076534322 23361610029905 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Rows 2..6 are A002620(n+2), A058389, A058390, A058391, A058392.

Cf. A304942, A306017.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
Table[RowSumMats[n-k, n-k, k], {n, 1, 11}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)

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, k)={polcoeff(1/prod(j=1, #q, my(g=gcd(t, q[j])); (1 - x^(q[j]/g) + O(x*x^k))^g), k)}
RowSumMats(n, m, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoeff(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n)); s/m!}
for(n=1, 8, for(k=0, 6, print1(RowSumMats(n, n, k), ", ")); print)

A326783 BII-numbers of uniform set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 20, 32, 36, 48, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 260, 272, 276, 288, 292, 304, 308, 512, 516, 528, 532, 544, 548, 560, 564, 768, 772, 784, 788, 800, 804, 816, 820, 1024, 1088, 2048, 2052, 2064, 2068, 2080

Offset: 1

Gus Wiseman, Jul 25 2019

nonn

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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. A set-system is uniform if all edges have the same size.

Alternatively, these are numbers whose binary indices all have the same binary weight, where the binary weight of a nonnegative integer is the numbers of 1's in its binary digits.

			The sequence of all uniform set-systems together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}

Cf. A000120, A029931, A048793, A070939, A047966, A306017, A306021, A319269, A320324, A326031, A326784 (regular), A326785 (uniform regular), A326788.

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

A306319 Number of length-rectangular twice-partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 60, 82, 131, 177, 286, 376, 582, 793, 1202, 1610, 2450, 3274, 4906, 6665, 9770, 13274, 19690, 26506, 38596, 53006, 76432, 104189, 150844, 205282, 294304, 404146, 573140, 786169, 1119457, 1527554, 2155953, 2965567, 4163955, 5701816

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is length-rectangular if all parts have the same number of parts.

			The a(5) = 14 length-rectangular twice-partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Dominates A319066 (rectangular partitions of partitions), which dominates A323429 (rectangular plane partitions).

Cf. A000219, A001970, A063834 (twice-partitions), A089299, A271619, A279787 (sum-rectangular twice-partitions), A305551, A306017, A306318 (square case), A323531.

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],SameQ@@Length/@#&],{ptn,IntegerPartitions[n]}]],{n,20}]

A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 5, 23, 21, 5, 1, 1, 7, 79, 162, 66, 7, 1, 1, 11, 274, 1636, 1338, 192, 11, 1, 1, 15, 1003, 19977, 43686, 10585, 565, 15, 1, 1, 22, 3763, 298416, 2142277, 1178221, 82694, 1579, 22, 1, 1, 30, 14723, 5300296, 149056260, 232984145, 30370346, 612700, 4348, 30, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.

			Array begins:
============================================================
n\k | 0  1   2     3        4           5              6
----+-------------------------------------------------------
  0 | 1  1   1     1        1           1              1 ...
  1 | 1  1   2     3        5           7             11 ...
  2 | 1  2   7    23       79         274           1003 ...
  3 | 1  3  21   162     1636       19977         298416 ...
  4 | 1  5  66  1338    43686     2142277      149056260 ...
  5 | 1  7 192 10585  1178221   232984145    74676589469 ...
  6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...
  ...
The A(2,2) = 7 matrices are:
  [1 0]  [2 0]  [1 1]  [2 1]  [2 0]  [1 1]  [2 2]
  [1 0]  [0 1]  [1 0]  [0 1]  [0 2]  [1 1]
  [0 1]  [0 1]  [0 1]
  [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152

Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.
Columns k=0..3 are A000012, A000041, A331722, A331723.
Cf. A219727, A306017, A316674, A318951, A331315, A331461.

\\ See A318951 for RowSumMats
T(n, k)={RowSumMats(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

A306017(n) = Sum_{d|n} A(n/d, d).

A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 3, 16, 8, 14, 3, 39, 3, 16, 15, 40, 3, 50, 3, 54, 17, 20, 3, 135, 10, 22, 25, 73, 3, 129, 3, 119, 21, 26, 19, 273, 3, 28, 23, 217, 3, 203, 3, 123, 74, 32, 3, 590, 12, 106, 27, 154, 3, 370, 23, 343, 29, 38, 3, 963, 3, 40, 95, 450, 25, 467, 3

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

An unlabeled version of A279789.

			The a(1) = 1 through a(6) = 12 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (111)      (22)          (11111)          (33)
       (1)(1)  (1)(1)(1)  (1111)        (1)(1)(1)(1)(1)  (222)
                          (2)(2)                         (3)(3)
                          (2)(11)                        (111111)
                          (11)(11)                       (3)(111)
                          (1)(1)(1)(1)                   (2)(2)(2)
                                                         (111)(111)
                                                         (2)(2)(11)
                                                         (2)(11)(11)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(1)(1)(1)

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A001970, A006171 (constant parts), A007716, A034729, A047966 (uniform partitions), A047968, A279787, A279789 (twice-partitions version), A305551 (equal part-sums), A306017, A319056, A323766, A323775, A323776.

Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@ptn]],{ptn,Select[IntegerPartitions[n],SameQ@@#&]}]],{n,30}]

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019

a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.

A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

Original entry on oeis.org

1, 1, 2, 7, 12, 35, 111, 247, 624, 1843, 6717, 15020, 46847, 124808, 412577, 1658973, 4217546, 12997734, 40786810, 126971940, 437063393, 2106317043, 5499108365, 19037901867, 59939925812, 210338815573, 683526043801, 2741350650705, 14848209030691, 41533835240731, 151548411269815

Offset: 0

Gus Wiseman, Jul 13 2019

nonn

The number of non-isomorphic multiset partitions of weight n is A007716(n).

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}
                  {{1},{2,2}}  {{1,2,3,4}}
                  {{1},{2,3}}  {{1},{1,1,1}}
                  {{2},{1,2}}  {{1},{1,2,2}}
                               {{1},{2,2,2}}
                               {{1},{2,3,3}}
                               {{1},{2,3,4}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

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

Row sums of A332260.

Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}
a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Feb 08 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 08 2020

A301481 Number of unlabeled uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 2, 4, 12, 58, 2381, 14026281, 29284932065996445, 468863491068204425232922367150021, 1994324729204021501147398087008429476673379600542622915802043462326345

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) = 12 hypergraphs:
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{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}}

Alois P. Heinz, Table of n, a(n) for n = 0..14 (first 13 terms from Andrew Howroyd)

Row sums of A301922.

Cf. A003465, A038041, A055621, A298422, A301920, A306017-A306021.

\\ see A301922 for U(n,k).
a(n)={ if(n==0, 1, sum(k=1, n, U(n,k)-U(n-1,k))) } \\ Andrew Howroyd, Aug 10 2019

Terms a(6) and beyond from Andrew Howroyd, Aug 09 2019

cp's OEIS Frontend

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

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

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

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A318951 Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0).

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Mathematica

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A326783 BII-numbers of uniform set-systems.

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Mathematica

A306319 Number of length-rectangular twice-partitions of n.

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Mathematica

A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

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A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

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