A326535 -id:A326535 - cp's OEIS frontend

A382304 MM-numbers of multiset partitions into sets with a common sum.

1, 2, 3, 4, 5, 8, 9, 11, 13, 16, 17, 25, 27, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 81, 83, 101, 109, 113, 121, 125, 127, 128, 137, 139, 143, 149, 157, 163, 167, 169, 179, 181, 191, 199, 211, 233, 241, 243, 256, 257, 269, 271, 277, 283, 289, 293, 313, 317

Offset: 1

Gus Wiseman, Apr 01 2025

nonn

Also products of prime numbers of squarefree index with a common sum of prime indices.

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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Set partitions of this type are counted by A035470.
Twice-partitions of this type are counted by A279788.
For just strict blocks we have A302478.
For just a common sum we have A326534, distinct sums A326535.
Factorizations of this type are counted by A382080.
For distinct instead of equal sums we have A382201.
For constant instead of strict blocks we have A382215.
Normal multiset partitions of this type are counted by A382429.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A058891 counts set-systems, covering A003465, connected A323818.
Cf. A000720, A005117, A050320, A050326, A293511, A302242, A302494, A381635, A381636, A381995.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@UnsameQ@@@prix/@prix[#]&]

Equals A302478 /\ A326534.

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Gus Wiseman, Mar 26 2025

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

			The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
The complement is counted by A381717, ranks A381636, zeros of A381635.
For strict instead of constant blocks we have A381992, ranks A382075.
For a unique choice we have A382301, ranks A381991.
Normal multiset partitions of this type are counted by A382203, sets A381718.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]

A327903 Number of set-systems covering n vertices where every edge has a different sum.

Original entry on oeis.org

1, 1, 5, 77, 7369, 10561753, 839653402893, 15924566366443524837, 315320784127456186118309342769, 29238175285109256786706269143580213236526609, 59347643832090275881798554403880633753161146711444051797893301

Offset: 0

Gus Wiseman, Sep 30 2019

nonn

A set-system is a set of nonempty sets. It is covering if there are no isolated (uncovered) vertices.

			The a(3) = 77 set-systems:
  123  1-23    1-2-3      1-2-3-13      1-2-3-13-23     1-2-3-13-23-123
       2-13    1-2-13     1-2-3-23      1-2-12-13-23    1-2-12-13-23-123
       1-123   1-2-23     1-2-12-13     1-2-3-13-123
       12-13   1-3-23     1-2-12-23     1-2-3-23-123
       12-23   2-3-13     1-2-13-23     1-2-12-13-123
       13-23   1-12-13    1-2-3-123     1-2-12-23-123
       2-123   1-12-23    1-3-13-23     1-2-13-23-123
       3-123   1-13-23    2-3-13-23     1-3-13-23-123
       12-123  1-2-123    1-12-13-23    2-3-13-23-123
       13-123  1-3-123    1-2-12-123    1-12-13-23-123
       23-123  2-12-13    1-2-13-123    2-12-13-23-123
               2-12-23    1-2-23-123
               2-13-23    1-3-13-123
               2-3-123    1-3-23-123
               3-13-23    2-12-13-23
               1-12-123   2-3-13-123
               1-13-123   2-3-23-123
               12-13-23   1-12-13-123
               1-23-123   1-12-23-123
               2-12-123   1-13-23-123
               2-13-123   2-12-13-123
               2-23-123   2-12-23-123
               3-13-123   2-13-23-123
               3-23-123   3-13-23-123
               12-13-123  12-13-23-123
               12-23-123
               13-23-123

Andrew Howroyd, Table of n, a(n) for n = 0..20
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The antichain case is A326572.
The graphical case is A327904.
Cf. A003465, A006126, A035470, A275780, A321469, A326030, A326519, A326535, A326565, A326571, A326573.

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]]]];
qes[n_]:=Select[stableSets[Subsets[Range[n],{1,n}],Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[qes[n]],{n,0,4}]

\\ by inclusion/exclusion on covered vertices.
C(v)={my(u=Vecrev(-1 + prod(k=1, #v, 1 + x^v[k]))); prod(i=1, #u, 1 + u[i])}
a(n)={my(s=0); forsubset(n, v, s += (-1)^(n-#v)*C(v)); s} \\ Andrew Howroyd, Oct 02 2019

Terms a(4) and beyond from Andrew Howroyd, Oct 02 2019

A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

Original entry on oeis.org

1, 1, 2, 8, 48, 432, 5184, 82944, 1658880, 41472000, 1244160000, 44789760000, 1881169920000, 92177326080000, 5161930260480000, 330363536670720000, 23786174640291840000, 1926680145863639040000, 173401213127727513600000, 17340121312772751360000000

Offset: 0

Gus Wiseman, Sep 30 2019

nonn

			The graph with edge-set {{1,2},{1,3},{1,4},{2,3}}, which looks like a triangle with a tail, has edges {1,4} and {2,3} with equal sum, so is not counted under a(4).

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
Gus Wiseman, The a(4) = 48 graphs where every edge has a different sum.

The generalization to antichains is A326030.

Cf. A006125, A006129, A275780, A321469, A326519, A326535, A327903.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1)*ceil(n/2)*ceil(n/2+1/4))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 03 2019

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]]]];
qes[n_]:=stableSets[Subsets[Range[n],{2}],Total[#1]==Total[#2]&];
Table[Length[qes[n]],{n,0,5}]

a(n) = {prod(k=1, 2*n+1, ceil(k/4))} \\ Andrew Howroyd, Oct 02 2019

a(n) = Product_{k=1..2*n+1} ceiling(k/4). - Andrew Howroyd, Oct 02 2019

Terms a(8) and beyond from Andrew Howroyd, Oct 02 2019

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 5, 2, 4, 5, 12, 1, 2, 2, 5, 2, 5, 4, 13, 2, 4, 5, 13, 5, 13, 13, 43, 1, 2, 2, 5, 2, 5, 5, 13, 2, 5, 4, 14, 5, 13, 14, 42, 2, 4, 5, 13, 5, 14, 13, 43, 5, 13, 14, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13

Offset: 0

Gus Wiseman, Jul 12 2020

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.

			The a(n) set partitions for n = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version for twice-partitions is A271619.
The version for partitions of partitions is (also) A271619.
These set partitions are counted by A275780.
The version for factorizations is A321469.
The version for normal multiset partitions is A326519.
The version for equal block-sums is A336137.
Set partitions with distinct block-lengths are A007837.
Set partitions of binary indices are A050315.
Twice-partitions with equal sums are A279787.
Partitions of partitions with equal sums are A305551.
Normal multiset partitions with equal block-lengths are A317583.
Multiset partitions with distinct block-sums are ranked by A326535.
Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],UnsameQ@@Total/@#&]],{n,0,100}]

A382426 MM-numbers of sets of constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 77, 82, 83, 85, 93, 95, 97, 102, 103, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159, 161, 165, 166

Offset: 1

Gus Wiseman, Apr 01 2025

nonn

Also products of prime numbers of prime power index with distinct sums of prime indices.

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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}

Twice-partitions of this type are counted by A279786.
For just constant blocks we have A302492.
For just distinct sums we have A326535.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A382201.
Normal multiset partitions of this type are counted by A382203.
For equal instead of distinct sums we have A382215.
An opposite version is A382304.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A000961, A302242, A302494, A321469, A326534, A355743, A356065, A381636, A381716.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@prix/@prix[#]&&And@@SameQ@@@prix/@prix[#]&]

Equals A302492 /\ A326535.

cp's OEIS Frontend

A382304 MM-numbers of multiset partitions into sets with a common sum.

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A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

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A327903 Number of set-systems covering n vertices where every edge has a different sum.

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A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

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A336138 Number of set partitions of the binary indices of n with distinct block-sums.

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A382426 MM-numbers of sets of constant multisets with distinct sums.

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