A116539 -id:A116539 - cp's OEIS frontend

A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 47

Offset: 0

Gus Wiseman, Mar 31 2025

First differs from A382523 at a(12) = 47, A382523(12) = 45.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

			Differs from A382523 in counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
  {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292444.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.
For distinct sums we have A382523, complement A382430.
Normal multiset partitions: A034691, A035310, A116540, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n];
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[strnorm[n], Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]], {n,0,5}]

A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 8, 15, 9, 14, 15, 17, 13, 22, 14, 25, 21, 23, 19, 34, 24, 29, 28, 37, 27, 45, 29, 44, 38, 43, 43, 59, 40, 51, 48, 69, 48, 71, 52, 73, 69, 72, 61, 93, 72, 91, 77, 99, 78, 105, 95, 119, 95, 113, 96, 146, 107, 126, 123, 151, 130

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 7 multiset partitions:
  {1} {11}   {111}     {1111}       {11111}         {111111}
      {1}{1} {2}{11}   {11}{11}     {2}{11}{11}     {111}{111}
             {1}{1}{1} {2}{2}{11}   {2}{2}{2}{11}   {22}{1111}
                       {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
                                                    {2}{2}{11}{11}
                                                    {2}{2}{2}{2}{11}
                                                    {1}{1}{1}{1}{1}{1}
The a(1) = 1 through a(7) = 5 factorizations:
  2  4    8      16       32         64           128
     2*2  3*4    4*4      3*4*4      8*8          3*4*4*4
          2*2*2  3*3*4    3*3*3*4    9*16         3*3*3*4*4
                 2*2*2*2  2*2*2*2*2  4*4*4        3*3*3*3*3*4
                                     3*3*4*4      2*2*2*2*2*2*2
                                     3*3*3*3*4
                                     2*2*2*2*2*2

Christian Sievers, Table of n, a(n) for n = 0..25000

Without a common sum we have A055887.
Twice-partitions of this type are counted by A279789.
Without constant blocks we have A326518.
For distinct block-sums and strict blocks we have A381718.
Factorizations of this type are counted by A381995.
For distinct instead of equal block-sums we have A382203.
For strict instead of constant blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A255906 counts normal multiset partitions, row sums of A317532.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A304969, A356945.
Set multipartitions: A116540, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

h(s,x)=my(t=0,p=1,k=1);while(s%k==0,p*=1/(1-x^(s/k))-1;t+=p;k+=1);t
lista(n)=Vec(1+sum(s=1,n,h(s,x+O(x*x^n)))) \\ Christian Sievers, Apr 05 2025

G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - Christian Sievers, Apr 05 2025

Terms a(16) and beyond from Christian Sievers, Apr 04 2025

A382429 Number of normal multiset partitions of weight n into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 26, 57, 113, 283, 854, 2401, 6998, 24072, 85061, 308956, 1190518, 4770078, 19949106, 87059592

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 13 partitions:
  {1} {12}   {123}     {1234}       {12345}         {123456}
      {1}{1} {3}{12}   {12}{12}     {24}{123}       {123}{123}
             {1}{1}{1} {14}{23}     {34}{124}       {125}{134}
                       {3}{3}{12}   {3}{12}{12}     {135}{234}
                       {1}{1}{1}{1} {5}{14}{23}     {145}{235}
                                    {3}{3}{3}{12}   {12}{12}{12}
                                    {1}{1}{1}{1}{1} {14}{14}{23}
                                                    {14}{23}{23}
                                                    {16}{25}{34}
                                                    {3}{3}{12}{12}
                                                    {5}{5}{14}{23}
                                                    {3}{3}{3}{3}{12}
                                                    {1}{1}{1}{1}{1}{1}
The corresponding factorizations:
  2  6    30     210      2310       30030
     2*2  5*6    6*6      21*30      30*30
          2*2*2  14*15    35*42      6*6*6
                 5*5*6    5*6*6      66*70
                 2*2*2*2  5*5*5*6    110*105
                          11*14*15   154*165
                          2*2*2*2*2  5*5*6*6
                                     14*14*15
                                     14*15*15
                                     26*33*35
                                     5*5*5*5*6
                                     11*11*14*15
                                     2*2*2*2*2*2

Without the common sum we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279788.
For common sizes instead of sums we have A317583.
Without strict blocks we have A326518, non-strict blocks A326517.
For a common length instead of sum we have A331638.
For distinct instead of equal block-sums we have A381718.
Factorizations of this type are counted by A382080.
For distinct block-sums and constant blocks we have A382203.
For constant instead of strict blocks we have A382204.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A255906, A304969, A317532.
Set multipartitions: A089259, A116539, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A000110, A034691, A038041, A050320, A255903, A326520, A381633, A381996, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(11) from Robert Price, Mar 30 2025

a(12)-a(20) from Christian Sievers, Apr 06 2025

A326914 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 5, 1, 12, 15, 18, 64, 52, 20, 166, 340, 203, 18, 332, 1315, 1866, 877, 15, 566, 3895, 9930, 10710, 4140, 11, 864, 9770, 39960, 74438, 64520, 21147, 6, 1214, 21848, 134871, 386589, 564508, 408096, 115975, 3, 1596, 44880, 402756, 1668338, 3652712

Offset: 0

Alois P. Heinz, Sep 13 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1, 12,   15;
           18,   64,    52;
           20,  166,   340,    203;
           18,  332,  1315,   1866,    877;
           15,  566,  3895,   9930,  10710,   4140;
           11,  864,  9770,  39960,  74438,  64520,  21147;
            6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A000110.
Row sums give A116539.
Column sums give A003465.
Cf. A001787, A255903, A326962 (this triangle read by columns), A327115, A327116, A327117.

C:= binomial:
g:= proc(n) option remember; n*2^(n-1) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);

c = Binomial;
g[n_] := g[n] = n*2^(n - 1);
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327115(n).
T(n*2^(n-1),n) = T(A001787(n),n) = 1.
T(n*2^(n-1)-1,n) = n for n >= 2.

A326962 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

Original entry on oeis.org

1, 1, 2, 2, 1, 5, 12, 18, 20, 18, 15, 11, 6, 3, 1, 15, 64, 166, 332, 566, 864, 1214, 1596, 1975, 2320, 2600, 2780, 2842, 2780, 2600, 2320, 1979, 1608, 1238, 908, 626, 404, 246, 136, 69, 32, 12, 4, 1, 52, 340, 1315, 3895, 9770, 21848, 44880, 86275, 157140

Offset: 0

Alois P. Heinz, Sep 13 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1, 12,   15;
           18,   64,    52;
           20,  166,   340,    203;
           18,  332,  1315,   1866,    877;
           15,  566,  3895,   9930,  10710,   4140;
           11,  864,  9770,  39960,  74438,  64520,  21147;
            6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
  ...

Alois P. Heinz, Columns k = 0..10, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A000110.
Row sums give A116539.
Column sums give A003465.
Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);

c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, k 2^(k-1)}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327115(n).
T(n*2^(n-1),n) = T(A001787(n),n) = 1.
T(n*2^(n-1)-1,n) = n for n >= 2.

A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9

Without distinct sums we have A055887.
Twice-partitions of this type are counted by A279786.
For distinct blocks instead of sums we have A304969.
Without constant blocks we have A326519.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A381718.
For equal instead of distinct block-sums we have A382204.
For equal block-sums and strict blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A116540, A255906, A317532.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(14)-a(26) from Christian Sievers, Apr 04 2025

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

Original entry on oeis.org

1, 1, 1, 6, 8, 35, 292, 673, 2818, 16956, 219772, 636748, 3768505, 20309534, 183403268, 3227600747, 12272598308, 81353466578, 561187259734, 4416808925866, 50303004612136, 1238783066956740, 5566249468690291, 44970939483601100, 330144217684933896, 3131452652308459402

Offset: 0

Gus Wiseman, Mar 29 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 8 multiset partitions:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
                  {{1},{1,2}}  {{1},{1,2,3}}
                  {{1},{2,3}}  {{1},{2,3,4}}
                  {{2},{1,2}}  {{2},{1,2,3}}
                  {{2},{1,3}}  {{2},{1,3,4}}
                  {{3},{1,2}}  {{3},{1,2,3}}
                               {{3},{1,2,4}}
                               {{4},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

For distinct sums instead of sizes we have A116539, see A050326.
Without distinct lengths we have A116540 (normal set multipartitions).
Without strict blocks we have A326517, for sum instead of size A326519.
For equal instead of distinct sizes we have A331638.
Twice-partitions of this type are counted by A358830.
For distinct sums instead of sizes we have A381718.
For equal instead of distinct sizes we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A279785, A317532, A326518, A381633, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Length/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Mar 31 2025

a(10) onwards from Andrew Howroyd, Mar 31 2025

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

A322698 Number of regular graphs with half-edges on n labeled vertices.

Original entry on oeis.org

1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610

Offset: 0

Gus Wiseman, Dec 23 2018

nonn

A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.

			The a(3) = 10 edge sets:
  {}
  {{1},{2,3}}
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Wikipedia, Regular graph

Row sums of A333157.

Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Union/@Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,6}]

for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(10)-a(18) from Andrew Howroyd, Aug 28 2019

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

cp's OEIS Frontend

A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

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A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

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A382429 Number of normal multiset partitions of weight n into sets with a common sum.

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A326914 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

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A326962 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

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A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

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A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

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A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.