A237258 -id:A237258 - cp's OEIS frontend

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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304

Offset: 1

N. J. A. Sloane

			Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
  (11)  (22)    (33)      (44)
        (211)   (321)     (422)
        (1111)  (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.

Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.

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

Mathematica

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]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *) primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s]; Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)

Python

from itertools import combinations_with_replacement from sympy.utilities.iterables import partitions def A002219(n): return 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

Formula

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).

a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022

Extensions

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

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

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Author

N. J. A. Sloane

Keywords

nonn
nice

Comments

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

Examples

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)

References

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

Links

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.

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

Python

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

Formula

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

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

Extensions

More terms from Don Reble, Nov 03 2001

More terms from Alois P. Heinz, Jul 10 2012

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset: 0

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Author

Gus Wiseman, Sep 16 2023

Keywords

nonn
tabl

Comments

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.

Rows are palindromic.

Are there only two zeros in the whole triangle?

Examples

Triangle begins: 1 1 1 1 0 1 2 1 1 2 2 1 0 1 2 3 1 1 1 1 3 4 2 2 1 2 2 4 5 2 2 2 2 2 2 5 6 3 2 3 1 3 2 3 6 8 3 3 4 3 3 4 3 3 8 Row n = 6 counts the following strict partitions: (6) (5,1) (4,2) (3,2,1) (4,2) (5,1) (6) (5,1) (3,2,1) (3,2,1) (3,2,1) (3,2,1) (5,1) (4,2) (4,2) (3,2,1) (3,2,1) Row n = 10 counts the following strict partitions: A 91 82 73 64 532 64 73 82 91 A 64 541 532 532 541 541 541 532 532 541 64 73 631 721 631 631 4321 631 631 721 631 73 82 721 4321 721 4321 4321 721 4321 721 82 91 4321 4321 4321 4321 91 532 532 541 541 631 631 721 721 4321 4321

Links

Robert Price, Table of n, a(n) for n = 0..1325

Crossrefs

Columns k = 0 and k = n are A000009.

The non-strict complement is A046663, central column A006827.

Central column n = 2k is A237258.

For subsets instead of partitions we have A365381.

The non-strict case is A365543.

The complement is A365663.

A000124 counts distinct possible sums of subsets of {1..n}.

A364272 counts sum-full strict partitions, sum-free A364349.

Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Programs

Mathematica

Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#],k]&]], {n,0,10},{k,0,n}]

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

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Author

Gus Wiseman, Sep 17 2023

Keywords

nonn
tabl

Comments

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

Examples

Triangle begins: 1 1 1 1 2 1 2 2 2 2 2 2 3 2 2 3 3 3 3 3 3 3 4 3 5 3 4 3 5 5 4 5 5 4 5 5 5 6 5 6 7 6 5 6 5 7 7 7 7 7 7 7 7 7 7 8 9 8 8 8 11 8 8 8 9 8 Row n = 8 counts the following strict partitions: (8) (8) (8) (8) (8) (8) (8) (6,2) (7,1) (7,1) (7,1) (7,1) (7,1) (6,2) (5,3) (5,3) (6,2) (6,2) (6,2) (5,3) (5,3) (4,3,1) (5,3) (4,3,1) (5,2,1)

Links

Robert Price, Table of n, a(n) for n = 2..1226

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.

Crossrefs

Columns k = 0 and k = n are A025147.

The non-strict version is A046663, central column A006827.

Central column n = 2k is A321142.

The complement for subsets instead of strict partitions is A365381.

The complement is A365661, non-strict A365543, central column A237258.

Row sums are A365922.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A124506 appears to count combination-free subsets, differences of A326083.

A364272 counts sum-full strict partitions, sum-free A364349.

A364350 counts combination-free strict partitions, complement A364839.

Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448

Offset: 1

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Author

Gus Wiseman, Oct 26 2022

Keywords

nonn

Comments

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.

Examples

The terms together with their prime indices begin: 1: {} 4: {1,1} 9: {2,2} 12: {1,1,2} 16: {1,1,1,1} 25: {3,3} 30: {1,2,3} 36: {1,1,2,2} 40: {1,1,1,3} 48: {1,1,1,1,2} 49: {4,4} For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

The partitions with these Heinz numbers are counted by A002219.

A subset of A300061.

The squarefree case is A357854, counted by A237258.

Positions of nonzero terms in A357879.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.

Programs

Maple

filter:= proc(n) local F,s,t,i,R; F:= ifactors(n)[2]; F:= map(t -> [numtheory:-pi(t[1]),t[2]], F); s:= add(t[1]*t[2],t=F)/2; if not s::integer then return false fi; try R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20); catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false; end try; true end proc: filter(1):= true: select(filter, [$1..1000]); # Robert Israel, Oct 26 2023

Mathematica

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]; Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

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

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Author

Christian G. Bower, Oct 12 2001

Keywords

nonn

Comments

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.

Examples

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)

Crossrefs

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.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], MemberQ[Total/@Subsets[#],n]&]],{n,0,5}] (* Gus Wiseman, Apr 19 2024 *)

Extensions

More terms from Alois P. Heinz, Jun 12 2017

A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 3, 1, 7, 1, 11, 6, 4, 1, 15, 1, 22, 14, 5, 1, 30, 10, 1, 42, 25, 6, 1, 56, 1, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1, 231, 167, 55, 9, 1, 297, 1, 385, 278, 173, 28, 10, 1, 490, 1, 627, 480, 140, 91, 11, 1, 792, 343, 36, 1

Offset: 1

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Author

Gus Wiseman, Apr 14 2024

Keywords

nonn
tabf

Comments

These could be called d-quanimous partitions, cf. A002219, A064914, A321452.

Examples

Triangle begins: 1 2 1 3 1 5 3 1 7 1 11 6 4 1 15 1 22 14 5 1 30 10 1 42 25 6 1 56 1 77 53 30 15 7 1 101 1 135 89 8 1 176 65 21 1 Row n = 6 counts the following partitions: (6) (33) (222) (111111) (33) (321) (2211) (42) (2211) (21111) (51) (3111) (111111) (222) (21111) (321) (111111) (411) (2211) (3111) (21111) (111111)

Crossrefs

Row lengths are A000005.

Column k = 1 is A000041.

Inserting zeros gives A371954.

Row sums are A372121.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

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.

A371736 counts non-quanimous strict partitons, complement A371737.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

Cf. A006827, A027750, A035470, A064914, A321455, A365543, A371791, A371795.

Programs

Mathematica

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,1,8},{k,Divisors[n]}]

Extensions

More terms from Jinyuan Wang, Feb 13 2025

Name edited by Peter Munn, Mar 05 2025

A321142 Number of strict integer partitions of 2*n with no subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 15, 23, 30, 43, 57, 79, 102, 138, 174, 232, 292, 375, 471, 602, 741, 935, 1148, 1425, 1733, 2137, 2571, 3156, 3789, 4557, 5470, 6582, 7796, 9317, 11027, 13058, 15400, 18159, 21249, 24971, 29170, 33986, 39596, 46073, 53219, 61711, 71330, 82171

Offset: 0

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Author

Gus Wiseman, Oct 28 2018

Keywords

nonn

Examples

The a(1) = 1 through a(8) = 23 partitions: (2) (4) (6) (8) (10) (12) (14) (16) (3,1) (4,2) (5,3) (6,4) (7,5) (8,6) (9,7) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6) (7,1) (8,2) (9,3) (10,4) (11,5) (5,2,1) (9,1) (10,2) (11,3) (12,4) (6,3,1) (11,1) (12,2) (13,3) (7,2,1) (5,4,3) (13,1) (14,2) (7,3,2) (6,5,3) (15,1) (7,4,1) (8,4,2) (7,5,4) (8,3,1) (8,5,1) (7,6,3) (9,2,1) (9,3,2) (9,4,3) (9,4,1) (9,5,2) (10,3,1) (9,6,1) (11,2,1) (10,4,2) (8,3,2,1) (10,5,1) (11,3,2) (11,4,1) (12,3,1) (13,2,1) (6,5,4,1) (7,4,3,2) (9,4,2,1) (10,3,2,1)

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A002219, A006827, A108917, A237258, A275972, A279791, A284640, A305551.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!Or@@Table[SameQ[Total[#[[s]]],n/2],{s,Subsets[Range[Length[#]]]}]]&]],{n,2,20,2}]

Extensions

a(33)-a(48) from Giovanni Resta, Oct 30 2018

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

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Author

Gus Wiseman, Apr 07 2024

Keywords

nonn

Comments

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.

Examples

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)

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Extensions

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

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Author

David S. Newman, Nov 24 2013

Keywords

nonn

Comments

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

Examples

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)

References

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.

Links

Martin Fuller, C++ program

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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

Extensions

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

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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

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

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A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

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A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

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A064914 Number of ordered biquanimous partitions of 2n.

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A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

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