A006827 -id:A006827 - 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

A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 5, 3, 4, 4, 4, 4, 4, 4, 4, 7, 5, 7, 8, 7, 5, 7, 8, 7, 7, 8, 8, 7, 7, 8, 12, 9, 12, 9, 17, 9, 12, 9, 12, 14, 11, 12, 12, 13, 13, 12, 12, 11, 14, 21, 15, 19, 15, 21, 24, 21, 15, 19, 15, 21, 24, 19, 20, 19, 21, 22, 22, 21, 19, 20, 19, 24, 34, 23, 30, 24, 30, 25, 46, 25, 30, 24, 30, 23, 34

Offset: 2

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Author

N. J. A. Sloane

Keywords

nonn
easy
look
nice
tabl

Examples

For n = 4 there are two partitions (4, 2+2) with no subsum equal to 1, two (4, 3+1) with no subsum equal to 2 and two (4, 2+2) with no subsum equal to 3. Triangle T(n,k) begins: 1; 1, 1; 2, 2, 2; 2, 2, 2, 2; 4, 3, 5, 3, 4; 4, 4, 4, 4, 4, 4; 7, 5, 7, 8, 7, 5, 7; 8, 7, 7, 8, 8, 7, 7, 8; 12, 9, 12, 9, 17, 9, 12, 9, 12; ... From _Gus Wiseman_, Oct 11 2023: (Start) Row n = 8 counts the following partitions: (8) (8) (8) (8) (8) (8) (8) (62) (71) (71) (71) (71) (71) (62) (53) (53) (62) (62) (62) (53) (53) (44) (44) (611) (611) (611) (44) (44) (422) (431) (44) (53) (44) (431) (422) (332) (422) (521) (422) (332) (2222) (2222) (5111) (2222) (2222) (332) (End)

Links

Alois P. Heinz, Rows n = 2..98, flattened

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

Column k = 0 and diagonal k = n are both A002865.

Central diagonal n = 2k is A006827.

The complement with expanded domain is A365543.

The strict case is A365663, complement A365661.

Row sums are A365918, complement A304792.

For subsets instead of partitions we have A366320, complement A365381.

A000041 counts integer partitions, strict A000009.

A276024 counts distinct subset-sums of partitions.

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

Cf. A000124, A002219, A093971, A108917, A122768, A299701, A365658.

Programs

Maple

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`(0 in s or n in s, 0, `if`(n=0 or s={}, g(n, i), `if`(i<1, 0, b(n, i-1, s)+`if`(i>n, 0, b(n-i, i, select(y-> 0<=y and y<=n-i, map(x-> [x, x-i][], s))))))) end: T:= (n, k)-> b(n, n, {min(k, n-k)}): seq(seq(T(n, k), k=1..n-1), n=2..16); # Alois P. Heinz, Jul 13 2012

Mathematica

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i > 1, g[n, i-1], 0] + If[i > n, 0, g[n-i, i]]]; b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0 || s == {}, g[n, i], If[i < 1, 0, b[n, i-1, s] + If[i > n, 0, b[n-i, i, Select[Flatten[s /. x_ :> {x, x-i}], 0 <= # <= n-i &]]]]]]; t[n_, k_] := b[n, n, {Min[k, n-k]}]; Table[t[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 20 2013, translated from Maple *) Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#],k]&]],{n,2,10},{k,1,n-1}] (* Gus Wiseman, Oct 11 2023 *)

Extensions

Corrected and extended by Don Reble, Nov 04 2001

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}]

A237258 Number of strict partitions of 2n that include a partition of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 32, 43, 63, 84, 122, 158, 220, 293, 393, 511, 685, 881, 1156, 1485, 1925, 2445, 3147, 3952, 5019, 6323, 7924, 9862, 12336, 15259, 18900, 23294, 28646, 35091, 42985, 52341, 63694, 77336, 93588, 112973, 136367, 163874, 196638

Offset: 0

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Author

Clark Kimberling, Feb 05 2014

Keywords

nonn

Comments

A strict partition is a partition into distinct parts.

Examples

a(5) counts these partitions of 10: [5,4,1], [5,3,2], [4,3,2,1].

Links

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

Crossrefs

Cf. A000009, A237194, A235130.

The non-strict version is A002219, ranked by A357976.

These partitions are ranked by A357854.

A000712 counts distinct submultisets of partitions, strict A032302.

A304792 counts subset-sums of partitions, positive A276024, strict A284640.

Cf. A006827, A064914, A108917, A276107, A300061, A357879.

Programs

Mathematica

z = 24; Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[2 nn], # == DeleteDuplicates[#] &]]; Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], nn]] >= 1 &]]], {nn, z}] (* Peter J. C. Moses, Feb 04 2014 *)

Formula

a(n) = A237194(2n,n).

Extensions

a(31)-a(47) from Alois P. Heinz, Feb 07 2014

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

A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset: 0

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Author

Gus Wiseman, Sep 26 2023

Keywords

nonn

Comments

The complement (complete partitions) is A126796.

Examples

The a(0) = 0 through a(8) = 12 partitions: . . (2) (3) (4) (5) (6) (7) (8) (2,2) (3,2) (3,3) (4,3) (4,4) (3,1) (4,1) (4,2) (5,2) (5,3) (5,1) (6,1) (6,2) (2,2,2) (3,2,2) (7,1) (4,1,1) (3,3,1) (3,3,2) (5,1,1) (4,2,2) (4,3,1) (5,2,1) (6,1,1) (2,2,2,2) (5,1,1,1)

Links

Joerg Arndt, Table of n, a(n) for n = 0..10000

Crossrefs

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.

The complement is A126796, ranks A325781, strict A188431.

These partitions have ranks A365830.

The strict case is A365831.

Row sums of A365923 without the first column, strict A365545.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions w/o a submultiset summing to k, strict A365663.

A276024 counts positive subset-sums of partitions, strict A284640.

A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

Programs

Mathematica

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]]; Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

Formula

a(n) = A000041(n) - A126796(n).

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

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

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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A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

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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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A237258 Number of strict partitions of 2n that include a partition of n.

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

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