A371783 -id:A371783 - cp's OEIS frontend

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

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

A006827 Number of partitions of 2n with all subsums different from n.

Original entry on oeis.org

1, 2, 5, 8, 17, 24, 46, 64, 107, 147, 242, 302, 488, 629, 922, 1172, 1745, 2108, 3104, 3737, 5232, 6419, 8988, 10390, 14552, 17292, 23160, 27206, 36975, 41945, 57058, 65291, 85895, 99384, 130443, 145283, 193554, 218947, 281860, 316326, 413322, 454229, 594048

Offset: 1

N. J. A. Sloane

Partitions of this type are also called non-biquanimous partitions. - Gus Wiseman, Apr 19 2024

			From _Gus Wiseman_, Apr 19 2024: (Start)
The a(1) = 1 through a(5) = 17 partitions (A = 10):
  (2)  (4)   (6)    (8)     (A)
       (31)  (42)   (53)    (64)
             (51)   (62)    (73)
             (222)  (71)    (82)
             (411)  (332)   (91)
                    (521)   (433)
                    (611)   (442)
                    (5111)  (622)
                            (631)
                            (721)
                            (811)
                            (3331)
                            (4222)
                            (6211)
                            (7111)
                            (22222)
                            (61111)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

The complement is counted by A002219, ranks A357976.
Central diagonal of A046663.
The strict case is A321142, even bisection of A371794 (odd A078408).
This is the "bi-" version of A321451, ranks A321453.
Column k = 0 of A367094.
These partitions have Heinz numbers A371731.
Even bisection of A371795 (odd A058695).
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A237258, A305551, A321452, A365543, A365663, A366320, A371736, A371782, A371792.

b:= proc(n, i, s) option remember;
      `if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+
      `if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,
                 map(x-> [x, x-i][], s))), 0))))
    end:
a:= n-> b(2*n, 2*n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; a[n_] := b[2*n, 2*n, {n}]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)

from itertools import combinations_with_replacement
from collections import Counter
from sympy import npartitions
from sympy.utilities.iterables import partitions
def A006827(n): return npartitions(n<<1)-len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023

a(n) = A000041(2*n) - A002219(n).

a(n) = A046663(2*n,n).

More terms from Don Reble, Nov 03 2001

More terms from Alois P. Heinz, Jul 10 2012

A064914 Number of ordered biquanimous partitions of 2n.

Original entry on oeis.org

1, 1, 5, 23, 105, 449, 1902, 7828, 31976, 129200, 520425, 2088217, 8371186, 33514797, 134140430, 536699674, 2147154667, 8589198795, 34358341823, 137435830265, 549749857574, 2199010044813, 8796067657649, 35184315676573, 140737380485376, 562949713881526

Offset: 0

Christian G. Bower, Oct 12 2001

nonn

A biquanimous partition is one that can be bisected into two equal sized parts: e.g. 3+2+1 is a biquanimous partition of 6 as it contains 3 and 2+1, but 5+1 is not.

			From _Gus Wiseman_, Apr 19 2024: (Start)
The a(0) = 1 through a(3) = 23 biquanimous compositions:
  ()  (11)  (22)    (33)
            (112)   (123)
            (121)   (132)
            (211)   (213)
            (1111)  (231)
                    (312)
                    (321)
                    (1113)
                    (1122)
                    (1131)
                    (1212)
                    (1221)
                    (1311)
                    (2112)
                    (2121)
                    (2211)
                    (3111)
                    (11112)
                    (11121)
                    (11211)
                    (12111)
                    (21111)
                    (111111)
(End)

The unordered version (integer partitions) is A002219, ranks A357976.
The unordered complement is A371795, even case A006827, ranks A371731.
The complement is counted by A371956.
These compositions have ranks A372120, complement A372119.
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}] (* Gus Wiseman, Apr 19 2024 *)

More terms from Alois P. Heinz, Jun 12 2017

A371795 Number of non-biquanimous integer partitions of n.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922, 6842, 1172, 10143, 1745, 14883, 2108, 21637, 3104, 31185, 3737, 44583, 5232, 63261, 6419, 89134, 8988, 124754, 10390, 173525

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

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(8) = 8 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)    (7)        (8)
            (21)   (31)  (32)     (42)   (43)       (53)
            (111)        (41)     (51)   (52)       (62)
                         (221)    (222)  (61)       (71)
                         (311)    (411)  (322)      (332)
                         (2111)          (331)      (521)
                         (11111)         (421)      (611)
                                         (511)      (5111)
                                         (2221)
                                         (3211)
                                         (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)

The complement is counted by A002219 aerated, ranks A357976.
Even bisection is A006827, odd A058695.
The strict complement is A237258, ranks A357854.
This is the "bi-" version of A321451, ranks A321453.
The complement is the "bi-" version of A321452, ranks A321454.
These partitions have ranks A371731.
The strict case is A371794, bisections A321142, A078408.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n],Not@*biqQ]],{n,0,15}]

a(n) = if(n%2, numbpart(n), my(v=partitions(n/2), w=List([])); for(i=1, #v, for(j=1, i, listput(w, vecsort(concat(v[i], v[j]))))); numbpart(n)-#Set(w)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A232466 Number of dependent sets with largest element n.

Original entry on oeis.org

0, 0, 1, 2, 4, 10, 20, 44, 93, 198, 414, 864, 1788, 3687, 7541, 15382, 31200, 63191, 127482, 256857, 516404, 1037104, 2080357, 4170283, 8354078, 16728270, 33485553, 67012082, 134083661, 268249350, 536617010, 1073391040, 2147014212, 4294321453, 8589084469, 17178702571, 34358228044, 68717407217, 137436320023, 274874294012, 549751307200, 1099505394507, 2199015662477, 4398035921221, 8796080392378, 17592168222674

Offset: 1

David S. Newman, Nov 24 2013

nonn

Let S be a set of positive integers. If S can be divided into two subsets which have equal sums, then S is said to be a dependent set.

Dependent sets are also called biquanimous sets. Biquanimous partitions are counted by A002219 and ranked by A357976. - Gus Wiseman, Apr 18 2024

			From _Gus Wiseman_, Apr 18 2024: (Start)
The a(1) = 0 through a(6) = 10 sets:
  .  .  {1,2,3}  {1,3,4}    {1,4,5}    {1,5,6}
                 {1,2,3,4}  {2,3,5}    {2,4,6}
                            {1,2,4,5}  {1,2,3,6}
                            {2,3,4,5}  {1,2,5,6}
                                       {1,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,2,3,4,6}
                                       {1,2,4,5,6}
                                       {2,3,4,5,6}
(End)

J. Bourgain, Λ_p-sets in analysis: results, problems and related aspects. Handbook of the geometry of Banach spaces, Vol. I,195-232, North-Holland, Amsterdam, 2001.

Martin Fuller, C++ program

Cf. A161943, A232534.
Column k=2 of A248112.
First differences of A371791.
The complement is counted by A371793, differences of A371792.
This is the "bi-" case of A371797, differences of A371796.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
Cf. A035470, A064914, A321451, A321452, A366320, A367094, A371783, A371789.

b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),
      `if`(i*(i+1)/2 p+x^i,
       b(n+i, i-1) union b(abs(n-i), i-1))))
    end:
a:= n-> nops(b(n, n-1)):
seq(a(n), n=1..15);  # Alois P. Heinz, Nov 24 2013

b[n_, i_] := b[n, i] = If[i<1, If[n == 0, {0}, {}], If[i*(i+1)/2 < n, {}, b[n, i-1] ~Union~ Map[Function[p, p+x^i], b[n+i, i-1] ~Union~ b[Abs[n-i], i-1]]]]; a[n_] := Length[b[n, n-1]]; Table[Print[a[n]]; a[n], {n, 1, 24}] (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&biqQ[#]&]],{n,10}] (* Gus Wiseman, Apr 18 2024 *)

dep(S,k=0)=if(#S<2,return(if(#S,S[1],0)==k)); my(T=S[1..#S-1]);dep(T,abs(k-S[#S]))||dep(T,k+S[#S])
a(n)=my(S=[1..n-1]);sum(i=1,2^(n-1)-1,dep(vecextract(S,i),n)) \\ Charles R Greathouse IV, Nov 25 2013

a(n)=my(r=0);forsubset(n-1,s,my(t=sum(i=1,#s,s[i])+n);if(t%2==0,my(b=1);for(i=1,#s,b=bitor(b,b<Martin Fuller, Mar 21 2025

a(n) < 2^(n-2) because there are 2^(n-1) sets of which half have an even sum. - Martin Fuller, Mar 21 2025

a(9)-a(24) from Alois P. Heinz, Nov 24 2013
a(25) from Alois P. Heinz, Sep 30 2014
a(26) from Alois P. Heinz, Sep 17 2022
a(27) onwards from Martin Fuller, Mar 21 2025

A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 18, 38, 82, 175, 373, 787, 1651, 3439, 7126, 14667, 30049, 61249, 124440, 251922, 508779, 1025183, 2062287, 4142644, 8312927, 16667005, 33395275, 66880828, 133892910, 267976571, 536225921, 1072842931, 2146233971, 4293248183, 8587569636, 17176654105, 34355356676, 68713584720, 137430991937, 274867311960, 549741605972, 1099492913172, 2198998307679, 4398013970156, 8796049891377, 17592130283755, 35184298506429

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

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.

			For S = {1,3,4,6} we have {{1,6},{3,4}}, so S is counted under a(6).
The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}  {}       {}         {}         {}
              {1,2,3}  {1,2,3}    {1,2,3}    {1,2,3}
                       {1,3,4}    {1,3,4}    {1,3,4}
                       {1,2,3,4}  {1,4,5}    {1,4,5}
                                  {2,3,5}    {1,5,6}
                                  {1,2,3,4}  {2,3,5}
                                  {1,2,4,5}  {2,4,6}
                                  {2,3,4,5}  {1,2,3,4}
                                             {1,2,3,6}
                                             {1,2,4,5}
                                             {1,2,5,6}
                                             {1,3,4,6}
                                             {2,3,4,5}
                                             {2,3,5,6}
                                             {3,4,5,6}
                                             {1,2,3,4,6}
                                             {1,2,4,5,6}
                                             {2,3,4,5,6}

First differences are A232466.
The complement is counted by A371792, differences A371793.
This is the "bi-" case of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A357879, A365661, A366320, A365381, A365925, A367094, A371788, A371789.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],biqQ]],{n,0,15}]

a(16) onwards from Martin Fuller, Mar 21 2025

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 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

Offset: 1

Gus Wiseman, Apr 07 2024

nonn

These partitions are counted by A371795, even case A006827.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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.
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 aerated and ranked by A357976.

			The prime indices of 975 are {2,3,3,6}, which are not biquanimous, so 975 is in the sequence.
The prime indices of 900 are {1,1,2,2,3,3}, which can be partitioned into {{1,2,3},{1,2,3}} or {{3,3},{1,1,2,2}}, so 900 is not in the sequence.

The complement is A357976, counted by A002219.
For prime signature instead of indices we have A371782, complement A371781.
Partitions of this type are counted by A371795, even case A006827.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A002221, A027187, A035470, A064914, A299701, A300061, A318434, A321455, A326534, A357879, A366320, A367094.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Select[Range[100],Not@*biqQ@*prix]

Numbers n without a divisor d|n such that A056239(d) = A056239(n/d).

A371796 Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 19, 43, 94, 206, 439, 946, 1990, 4204, 8761, 18233, 37778, 78151, 160296, 328670, 670193, 1363543, 2772436, 5632801, 11404156, 23071507, 46613529, 94098106, 189959349, 383407198, 773009751

Offset: 0

Gus Wiseman, Apr 17 2024

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is counted under a(9).
The a(3) = 1 through a(6) = 19 subsets:
  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
           {1,3,4}    {1,3,4}      {1,3,4}
           {1,2,3,4}  {1,4,5}      {1,4,5}
                      {2,3,5}      {1,5,6}
                      {1,2,3,4}    {2,3,5}
                      {1,2,4,5}    {2,4,6}
                      {2,3,4,5}    {1,2,3,4}
                      {1,2,3,4,5}  {1,2,3,6}
                                   {1,2,4,5}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {2,3,4,5}
                                   {2,3,5,6}
                                   {3,4,5,6}
                                   {1,2,3,4,5}
                                   {1,2,3,4,6}
                                   {1,2,4,5,6}
                                   {2,3,4,5,6}
                                   {1,2,3,4,5,6}

The "bi-" version for integer partitions is A002219 aerated, ranks A357976.
The "bi-" version for strict partitions is A237258 aerated, ranks A357854.
The complement for integer partitions is A321451, ranks A321453.
The version for integer partitions is A321452, ranks A321454
The version for strict partitions is A371737, complement A371736.
The complement is counted by A371789, differences A371790.
The "bi-" version is A371791, complement A371792.
First differences are A371797.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
Cf. A000005, A018818, A035470, A038041, A232466, A279791, A321142, A365661, A371731, A371794, A371795.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,0,10}]

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

Original entry on oeis.org

0, 1, 3, 6, 12, 24, 46, 90, 174, 337, 651, 1261, 2445, 4753, 9258, 18101, 35487, 69823, 137704, 272366, 539797, 1071969, 2132017, 4245964, 8464289, 16887427, 33713589, 67336900, 134542546, 268894341, 537515903, 1074640717, 2148733325, 4296686409, 8592299548, 17183084263, 34364120060, 68725368752, 137446915007, 274888501928, 549770021804, 1099530342380, 2199048203425, 4398079052052, 8796136153039, 17592241805077, 35184445671235

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

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 subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
The a(0) = 0 through a(4) = 12 subsets:
  .  {1}  {1}    {1}    {1}
          {2}    {2}    {2}
          {1,2}  {3}    {3}
                 {1,2}  {4}
                 {1,3}  {1,2}
                 {2,3}  {1,3}
                        {1,4}
                        {2,3}
                        {2,4}
                        {3,4}
                        {1,2,4}
                        {2,3,4}

This is the "bi-" version of A371789, differences A371790.
The complement is counted by A371791, differences A232466.
First differences are A371793.
The complement is the "bi-" version of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],Not@*biqQ]],{n,0,10}]

a(16) onwards from Martin Fuller, Mar 21 2025

A371794 Number of non-biquanimous strict integer partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 27, 23, 38, 30, 54, 43, 76, 57, 104, 79, 142, 102, 192, 138, 256, 174, 340, 232, 448, 292, 585, 375, 760, 471, 982, 602, 1260, 741, 1610, 935, 2048, 1148, 2590, 1425, 3264, 1733, 4097, 2137, 5120, 2571, 6378

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

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(11) = 12 strict partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (521)  (81)   (91)   (92)
                                                  (432)  (631)  (A1)
                                                  (531)  (721)  (542)
                                                  (621)         (632)
                                                                (641)
                                                                (731)
                                                                (821)
                                                                (5321)

The complement is counted by A237258 aerated, ranks A357854.
Even bisection is A321142, odd A078408.
This is the "bi-" version of A371736, complement A371737.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!biqQ[#]&]],{n,0,30}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A006827 Number of partitions of 2n with all subsums different from n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A064914 Number of ordered biquanimous partitions of 2n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A371795 Number of non-biquanimous integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A232466 Number of dependent sets with largest element n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Views

Author

Keywords

Comments

Examples

Crossrefs