A006827 -id:A006827 - 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]&]

A321143 Number of non-isomorphic knapsack multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 10, 31, 87, 272, 835, 2673, 8805, 29583

Offset: 0

Gus Wiseman, Oct 28 2018

A multiset partition is knapsack if every distinct submultiset of the parts has a different multiset union.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 31 knapsack multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}
         {{1},{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},{1,1}}
                    {{1},{1},{1}}  {{1},{1,2,2}}
                    {{1},{2},{2}}  {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
Missing from this list are {{1},{1},{1,1}} and {{1},{2},{1,2}}, which are not knapsack.

Cf. A002219, A006827, A007716, A108917, A275972, A276024, A292886, A316983, A319616.

A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 25, 35, 53, 72, 108

Offset: 1

Gus Wiseman, Sep 30 2023

A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.

			The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
  {2}  {3}    {4}      {5}        {6}          {7}
       {1,3}  {1,4}    {1,5}      {1,6}        {1,7}
              {2,2}    {2,3}      {2,4}        {2,5}
              {1,1,4}  {1,1,5}    {3,3}        {3,4}
                       {1,2,5}    {1,1,6}      {1,1,7}
                       {1,1,1,5}  {1,2,6}      {1,2,7}
                                  {1,3,3}      {1,3,4}
                                  {2,2,2}      {2,2,3}
                                  {1,1,1,6}    {1,1,1,7}
                                  {1,1,2,6}    {1,1,2,7}
                                  {1,1,1,1,6}  {1,1,3,7}
                                               {1,2,2,7}
                                               {1,1,1,1,7}
                                               {1,1,1,2,7}
                                               {1,1,1,1,1,7}

For least instead of greatest we have A126796, ranks A325781, strict A188431.
These multisets have ranks A365830.
Counts appearances of n in the rank statistic A365920.
Column sums of A365921.
These multisets counted by sum are A365924, strict A365831.
The strict case is A366129.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365918 counts non-subset-sums of partitions.
A365923 counts partitions by non-subset sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365658, A365919, A365925.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

Table[Length[Subsets[Select[IntegerPartitions[n],UnsameQ@@#&],{2}]],{n,0,30}]

a(n) = binomial(A000009(n),2).

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&]

A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 20 2024

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.

			Triangle begins:
  1
  0  1
  0  2  1
  0  3  0  1
  0  5  3  0  1
  0  7  0  0  0  1
  0 11  6  4  0  0  1
  0 15  0  0  0  0  0  1
  0 22 14  0  5  0  0  0  1
  0 30  0 10  0  0  0  0  0  1
  0 42 25  0  0  6  0  0  0  0  1
  0 56  0  0  0  0  0  0  0  0  0  1
  0 77 53 30 15  0  7  0  0  0  0  0  1
Row n = 6 counts the following partitions:
  .  (6)       (33)      (222)     .  .  (111111)
     (51)      (321)     (2211)
     (42)      (3111)    (21111)
     (411)     (2211)    (111111)
     (33)      (21111)
     (321)     (111111)
     (3111)
     (222)
     (2211)
     (21111)
     (111111)

Row n has A000005(n) positive entries.
Column k = 1 is A000041.
Column k = 2 is A002219 (aerated), ranks A357976.
Column k = 3 is A002220 (aerated), ranks A371955.
Removing all zeros gives A371783.
Row sums are A372121.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, complement A371796.
Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, A371795.

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], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,0,10},{k,0,n}]

A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

Original entry on oeis.org

1, 3, 4, 9, 8, 22, 16, 42, 41, 74, 57, 183, 102, 233, 263, 463, 298, 875, 491, 1350, 1172, 1775, 1256, 4273, 2225, 4399, 4584, 8049, 4566, 14913, 6843, 18539, 15831, 22894, 18196, 53323, 21638, 48947, 50281, 94500, 44584, 144976, 63262, 173436, 169361, 202153

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums. The triangles A371783 and A371954 count k-quanimous partitions.

Row sums of A371783.
Row sums of A371954.
A000005 counts divisors.
A000041 counts integer partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A321452 counts quanimous partitions, complement A321451.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A035470, A064914, A321455, A365543, A371737, A371791, A371795.

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[Sum[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{k,Divisors[n]}],{n,1,10}]

T(n, d) = my(v=partitions(n/d), w=List([])); forvec(s=vector(d, i, [1, #v]), listput(w, vecsort(concat(vector(d, i, v[s[i]])))), 1); #Set(w);
a(n) = sumdiv(n, d, T(n, d)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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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A321143 Number of non-isomorphic knapsack multiset partitions of weight n.

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A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

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A366132 Number of unordered pairs of distinct strict integer partitions of n.

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

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

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A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

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