A035470 -id:A035470 - cp's OEIS frontend

A371956 Number of non-biquanimous compositions of 2n.

0, 1, 3, 9, 23, 63, 146, 364

Offset: 0

Gus Wiseman, Apr 20 2024

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The a(1) = 1 through a(3) = 9 compositions:
  (2)  (4)    (6)
       (1,3)  (1,5)
       (3,1)  (2,4)
              (4,2)
              (5,1)
              (1,1,4)
              (1,4,1)
              (2,2,2)
              (4,1,1)

The unordered complement is A002219, ranks A357976.
The unordered version is A006827, even case of A371795, ranks A371731.
The complement is counted by A064914.
These compositions have ranks A372119, complement A372120.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], !MemberQ[Total/@Subsets[#],n]&]],{n,0,5}]

A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The terms and corresponding compositions begin:
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  12: (1,3)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)

The unordered complement is A357976, counted by A002219.
The unordered version is A371731, counted by A371795, even case A006827.
These compositions are counted by A371956.
The complement is A372120, counted by A064914.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

A372120 Numbers k such that the k-th composition in standard order is biquanimous.

Original entry on oeis.org

0, 3, 10, 11, 13, 14, 15, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 136, 137, 138, 139, 140, 141, 142, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 165, 166, 167, 168, 169

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The terms and corresponding compositions begin:
   0: ()
   3: (1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  36: (3,3)
  37: (3,2,1)
  38: (3,1,2)
  39: (3,1,1,1)
  41: (2,3,1)
  43: (2,2,1,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  50: (1,3,2)
  51: (1,3,1,1)
  52: (1,2,3)
  53: (1,2,2,1)
  54: (1,2,1,2)

These compositions are counted by A064914.
The unordered version (integer partitions) is A357976, counted by A002219.
The unordered complement is A371731, counted by A371795, even case A006827.
The complement is A372119, counted by A371956.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

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

Original entry on oeis.org

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.

A275781 Number of set partitions of [n] having at least two blocks with equal element sums.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 43, 209, 1042, 5682, 32875, 200919, 1299092, 8848623, 63108778, 471509591, 3671980053, 29797471739, 251120234182, 2196428139810, 19882638695048, 186205573812059, 1799766634300161, 17946783669353462, 184277407872729741, 1947501874827169041

Offset: 0

Alois P. Heinz, Aug 08 2016

nonn

			a(3) = 1: 12|3
a(4) = 3: 12|3|4, 13|2|4, 14|23.
a(5) = 9: 12|3|45, 12|3|4|5, 13|25|4, 13|2|4|5, 14|23|5, 1|23|4|5, 14|2|3|5, 15|24|3, 1|25|34.

Cf. A000110, A035470, A275780.

a(n) = A000110(n) - A275780(n).

a(17)-a(25) from Christian Sievers, Aug 20 2024

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

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}]

A361910 Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 7, 12, 47, 99, 430, 1379, 5613, 21416, 127303, 532201, 3133846, 18776715, 114275757, 737859014

Offset: 1

Gus Wiseman, Apr 14 2023

Since (n+1)/2 is the mean of {1..n}, this sequence counts a type of "transitive" set partitions.

			The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
                   {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
                                {{14}{23}}      {{15}{234}}
                                {{1}{23}{4}}    {{1}{234}{5}}
                                {{14}{2}{3}}    {{12}{3}{45}}
                                {{1}{2}{3}{4}}  {{135}{2}{4}}
                                                {{14}{25}{3}}
                                                {{15}{24}{3}}
                                                {{1}{24}{3}{5}}
                                                {{15}{2}{3}{4}}
                                                {{1}{2}{3}{4}{5}}
The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).
The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(5).

For median instead of mean we have A361863.
A000110 counts set partitions.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, A000975 with integer median.
A327481 counts subsets by mean, A013580 by median.
A361865 counts set partitions with integer mean of means.
A361911 counts set partitions with integer sum of means.
Cf. A007837, A035470, A038041, A067538, A275714, A275780, A326512, A326513, A361864, A361866.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Mean[Join@@#]==Mean[Mean/@#]&]],{n,8}]

a(13)-a(17) from Christian Sievers, May 12 2025

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 48, 50, 52, 53, 55, 56, 57, 58, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92

Offset: 1

Gus Wiseman, Apr 14 2024

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 binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is not in the sequence.
The terms together with their binary expansions and binary indices begin:
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  10:  1010 ~ {2,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  23: 10111 ~ {1,2,3,5}

Set partitions with all equal block-sums are counted by A035470.
Positions of 1's in A336137 and A371735.
The complement is A371784.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[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,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]==1&]

A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.

Original entry on oeis.org

7, 13, 15, 22, 25, 27, 30, 31, 39, 42, 45, 47, 49, 51, 54, 59, 60, 62, 63, 75, 76, 82, 85, 87, 90, 93, 94, 95, 97, 99, 102, 107, 108, 109, 110, 115, 117, 119, 120, 122, 125, 126, 127, 141, 143, 147, 148, 153, 155, 158, 162, 165, 167, 170, 173, 175, 179, 180

Offset: 1

Gus Wiseman, Apr 16 2024

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 binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is in the sequence.
The terms together with their binary expansions and binary indices begin:
   7:     111 ~ {1,2,3}
  13:    1101 ~ {1,3,4}
  15:    1111 ~ {1,2,3,4}
  22:   10110 ~ {2,3,5}
  25:   11001 ~ {1,4,5}
  27:   11011 ~ {1,2,4,5}
  30:   11110 ~ {2,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  39:  100111 ~ {1,2,3,6}
  42:  101010 ~ {2,4,6}
  45:  101101 ~ {1,3,4,6}
  47:  101111 ~ {1,2,3,4,6}
  49:  110001 ~ {1,5,6}
  51:  110011 ~ {1,2,5,6}
  54:  110110 ~ {2,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  60:  111100 ~ {3,4,5,6}
  62:  111110 ~ {2,3,4,5,6}
  63:  111111 ~ {1,2,3,4,5,6}

Set partitions with all equal block-sums are counted by A035470.
Positions of terms > 1 in A336137 and A371735.
The complement is A371738.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[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,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]>1&]

cp's OEIS Frontend

A371956 Number of non-biquanimous compositions of 2n.

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A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

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A372120 Numbers k such that the k-th composition in standard order is biquanimous.

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

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A275781 Number of set partitions of [n] having at least two blocks with equal element sums.

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A327903 Number of set-systems covering n vertices where 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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A361910 Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.

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Extensions

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.