A306017 -id:A306017 - cp's OEIS frontend

A305551 Number of partitions of partitions of n where all partitions have the same sum.

1, 1, 3, 4, 9, 8, 22, 16, 43, 41, 77, 57, 201, 102, 264, 282, 564, 298, 1175, 491, 1878, 1509, 2611, 1256, 7872, 2421, 7602, 8026, 16304, 4566, 38434, 6843, 48308, 41078, 56582, 28228, 221115, 21638, 146331, 208142, 453017, 44584, 844773, 63262, 1034193, 1296708

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

			The a(4) = 9 partitions of partitions where all partitions have the same sum:
(4), (31), (22), (211), (1111),
(2)(2), (2)(11), (11)(11),
(1)(1)(1)(1).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000005, A001970, A001315, A007716, A038041, A055887, A063834, A271619, A289078, A298422, A306017.

Table[Sum[Binomial[PartitionsP[n/k]+k-1,k],{k,Divisors[n]}],{n,60}]

a(n)={if(n<1, n==0, sumdiv(n, d, binomial(numbpart(n/d) + d - 1, d)))} \\ Andrew Howroyd, Jun 22 2018

a(n) = Sum_{d|n} binomial(A000041(n/d) + d - 1, d).

A319056 Number of non-isomorphic multiset partitions of weight n in which (1) all parts have the same size and (2) each vertex appears the same number of times.

Original entry on oeis.org

1, 1, 4, 4, 10, 4, 21, 4, 26, 13, 28, 4, 128, 4, 39, 84, 150, 4, 358, 4, 956, 513, 86, 4, 12549, 1864, 134, 9582, 52366, 4, 301086, 4, 1042038, 407140, 336, 4690369, 61738312, 4, 532, 28011397, 2674943885, 4, 819150246, 4, 54904825372, 65666759973, 1303, 4, 4319823776760

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

a(p) = 4 for p prime. - Charlie Neder, Oct 15 2018

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 21 multiset partitions:
  (1)  (11)    (111)      (1111)        (11111)          (111111)
       (12)    (123)      (1122)        (12345)          (111222)
       (1)(1)  (1)(1)(1)  (1234)        (1)(1)(1)(1)(1)  (112233)
       (1)(2)  (1)(2)(3)  (11)(11)      (1)(2)(3)(4)(5)  (123456)
                          (11)(22)                       (111)(111)
                          (12)(12)                       (111)(222)
                          (12)(34)                       (112)(122)
                          (1)(1)(1)(1)                   (112)(233)
                          (1)(1)(2)(2)                   (123)(123)
                          (1)(2)(3)(4)                   (123)(456)
                                                         (11)(11)(11)
                                                         (11)(12)(22)
                                                         (11)(22)(33)
                                                         (11)(23)(23)
                                                         (12)(12)(12)
                                                         (12)(13)(23)
                                                         (12)(34)(56)
                                                         (1)(1)(1)(1)(1)(1)
                                                         (1)(1)(1)(2)(2)(2)
                                                         (1)(1)(2)(2)(3)(3)
                                                         (1)(2)(3)(4)(5)(6)

Andrew Howroyd, Table of n, a(n) for n = 0..75

Row sums of A322789.

Cf. A007716, A049311, A064573, A279787, A283877, A306017, A316980, A316981, A316983, A319616.

Terms a(12) and beyond from Andrew Howroyd, Feb 03 2022

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

Original entry on oeis.org

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

A319066 Number of partitions of integer partitions of n where all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 59, 82, 128, 176, 273, 371, 553, 768, 1119, 1544, 2235, 3084, 4410, 6111, 8649, 11982, 16901, 23383, 32780, 45396, 63365, 87622, 121946, 168407, 233605, 322269, 445723, 613922, 847131, 1164819, 1603431, 2201370, 3023660, 4144124, 5680816

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{1,1},{1,1}}      {{1,1,1,2}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001970, A047968, A261049, A279787, A305551, A306017, A319056.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Length/@#&]],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k*y + O(x*x^n))); concat([1], sum(k=1, n, EulerT(Vec(polcoef(p, k, y), -n))))} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 7, 1, 14, 10, 26, 1, 55, 1, 90, 68, 167, 1, 292, 1, 482, 345, 761, 1, 1291, 266, 1949, 1518, 3091, 1, 4793, 1, 7177, 5612, 10566, 2623, 16007, 1, 22912, 18992, 33619, 1, 48529, 1, 68758, 59187, 96571, 1, 137489, 11418, 189979, 167502, 264299

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

a(n) = 1 if and only if n is prime. - Chai Wah Wu, Nov 12 2018

			The a(2) = 1 through a(9) = 10 partitions:
  (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)        (333)
               (211)            (222)                (422)       (3321)
               (1111)           (321)                (431)       (32211)
                                (2211)               (2222)      (33111)
                                (3111)               (3221)      (222111)
                                (21111)              (3311)      (321111)
                                (111111)             (4211)      (2211111)
                                                     (22211)     (3111111)
                                                     (32111)     (21111111)
                                                     (41111)     (111111111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The partition (32111) can be partitioned as ((13)(112)), and the blocks both sum to 4, so (32111) is counted under a(8).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321451, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]>1&]],{n,10}]

a(n) = A000041(n) - A321451(n).

a(26)-a(52) from Alois P. Heinz, Nov 11 2018

A295193 Number of regular simple graphs on n labeled nodes.

Original entry on oeis.org

1, 2, 2, 8, 14, 172, 932, 45936, 1084414, 155862512, 10382960972, 6939278572096, 2203360500122300, 4186526756621772344, 3747344008241368443820, 35041787059691023579970848, 156277111373303386104606663422, 4142122641757598618318165240180096

Offset: 1

Álvar Ibeas, Nov 16 2017

nonn

			From _Gus Wiseman_, Dec 19 2018: (Start)
A graph is regular if all vertices have the same degree. For example, the a(4) = 8 simple regular graphs are:
  1 2
  3 4
.
  4---1  3---1  2---1
  3---2  4---2  4---3
.
  3---4  4---3  4---2
  |   |  |   |  |   |
  1---2  1---2  1---3
.
  4---3
  | X |
  2---1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..24
E. A. Bender and E. R. Canfield, The asymptotic number of labeled graphs with given degree sequences, Journal of Combinatorial Theory, Series A, 24 (1978), 296-307.
Andrew Howroyd, PARI Program
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
B. D. McKay, Applications of a technique for labelled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Wikipedia, Regular graph

Row sums of A059441.

Cf. A005176 (unlabeled equivalent), A058891, A116539, A299353, A306017, A306021, A319189, A319190, A319612, A319729.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,9}] (* Gus Wiseman, Dec 19 2018 *)

\\ See link for program file.
for(n=1, 10, print1(A295193(n), ", ")) \\ Andrew Howroyd, Aug 28 2019

a(16)-a(18) from Andrew Howroyd, Aug 28 2019

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 85, 89, 91, 93, 97, 99, 101, 103, 107, 109, 113, 121, 123, 125, 127, 128, 131, 135, 137, 139, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The terms together with their corresponding multiset multisystems (A302242):
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001222, A038041, A112798, A302242, A306017, A317583, A319066, A319169, A320325, A322794, A326533, A326534, A326535, A326536, A326537.

Select[Range[100],SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

is(n) = #Set(apply(p -> bigomega(primepi(p)), factor(n)[,1]~))<=1 \\ Rémy Sigrist, Oct 11 2018

A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 4, 14, 8, 20, 16, 55, 22, 100, 45, 108, 64, 296, 93, 489, 145, 447, 241, 1254, 284, 1692, 487, 1492, 627, 4564, 811, 6841, 1172, 4531, 1744, 12260, 1970, 21636, 3103, 12193, 3719, 44582, 4645, 63260, 6417, 29947, 8987, 124753, 9784, 162107, 14247

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

			The a(1) = 1 through a(9) = 20 partitions:
  (1)  (2)  (3)   (4)   (5)     (6)    (7)       (8)     (9)
            (21)  (31)  (32)    (42)   (43)      (53)    (54)
                        (41)    (51)   (52)      (62)    (63)
                        (221)   (411)  (61)      (71)    (72)
                        (311)          (322)     (332)   (81)
                        (2111)         (331)     (521)   (432)
                                       (421)     (611)   (441)
                                       (511)     (5111)  (522)
                                       (2221)            (531)
                                       (3211)            (621)
                                       (4111)            (711)
                                       (22111)           (3222)
                                       (31111)           (4221)
                                       (211111)          (4311)
                                                         (5211)
                                                         (6111)
                                                         (22221)
                                                         (42111)
                                                         (51111)
                                                         (411111)
A complete list of all multiset partitions of the partition (2111) into two or more blocks is: ((1)(112)), ((2)(111)), ((11)(12)), ((1)(1)(12)), ((1)(2)(11)), ((1)(1)(1)(2)). None of these has equal block-sums, so (2111) is counted toward a(5).
On the other hand, the partition (321) can be partitioned as ((12)(3)), which has two or more blocks and equal block-sums, so (321) is not counted toward a(6).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]==1&]],{n,10}]

a(n) = A000041(n) - A321452(n).

a(33)-a(50) from Alois P. Heinz, Nov 11 2018

A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.

Original entry on oeis.org

4, 8, 9, 12, 16, 25, 27, 30, 32, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 125, 128, 144, 150, 154, 160, 165, 169, 180, 192, 196, 198, 200, 210, 216, 220, 225, 240, 243, 252, 256, 264, 270, 273, 280, 286, 288, 289, 300, 320, 324, 325

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also Heinz numbers of integer partitions that can be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

			The sequence of all integer partitions that can be partitioned into two or more blocks with equal sums begins: (11), (111), (22), (211), (1111), (33), (222), (321), (11111), (2211), (3111), (21111), (44), (422), (111111), (431), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (333), (1111111), (221111), (3321), (541), (311111), (532), (66), (32211), (2111111), (4411), (5221), (33111).
The Heinz number of (32111) is 120, which has factorization (10*12) corresponding to the multiset partition ((13)(112)) whose blocks have equal sums, so 120 belongs to the sequence.

Positions of terms > 1 in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321453.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]!={}&]

A321453 Numbers that cannot be factored into two or more factors all having the same sum of prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 85

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also Heinz numbers of integer partitions that cannot be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

			The sequence of all integer partitions that cannot be partitioned into two or more blocks with equal sums begins: (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (221), (8), (311), (42), (51), (9), (2111), (61), (411).

Positions of 1's in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321454.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]=={}&]

cp's OEIS Frontend

A305551 Number of partitions of partitions of n where all partitions have the same sum.

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A319056 Number of non-isomorphic multiset partitions of weight n in which (1) all parts have the same size and (2) each vertex appears the same number of times.

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

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A319066 Number of partitions of integer partitions of n where all parts have the same length.

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A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

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A295193 Number of regular simple graphs on n labeled nodes.

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A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

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A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

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