A006827 -id:A006827 - cp's OEIS frontend

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3

Offset: 0

Gus Wiseman, Nov 07 2023

			The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
    0   1
    1   1
    2   2   1
    5   3   3
    8   4   9   1
   17   6  16   1   2
   24   7  33   4   9
   46  11  52   3  18   1   4
   64  12  91   6  38   3  15   1   1
  107  17 138   9  68   2  28   2  12   0   2
  147  19 219  12 117   6  56   3  34   2   9   0   3
Row n = 4 counts the following partitions:
  (8)     (44)        (431)      (221111)
  (71)    (3311)      (422)
  (62)    (2222)      (4211)
  (611)   (11111111)  (41111)
  (53)                (3221)
  (521)               (32111)
  (5111)              (311111)
  (332)               (22211)
                      (2111111)

Row sums w/o the first column are A002219, ranks A357976, strict A237258.
Column k = 0 is A006827.
Row sums are A058696.
Column k = 1 is A108917.
The corresponding rank statistic is A357879 (without empty rows).
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, ranks A366321.
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sums of partitions, rank statistic A299701.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]

T(n,1) = A108917(n).

A371782 Numbers with non-biquanimous prime signature.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 102

Offset: 1

Gus Wiseman, Apr 09 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 (aerated) and ranked by A357976.

Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.

			The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

A number's prime signature is given by A124010.
The complement for prime indices is A357976, counted by A002219 aerated.
For prime indices we have A371731, counted by A371795, even case A006827.
The complement is A371781, counted by A371839.
Partitions of this type are counted by A371840.
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.
A371792 counts non-biquanimous sets, complement A371791.
Cf. A035470, A064914, A232466, A299729, A367094, A371736, A371783, A371789.
Subsequence of A026424.

g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100],g[#]=={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; OddQ[sum] || CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] == 0]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Jul 24 2024 *)

A366754 Number of non-knapsack integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 52, 60, 94, 118, 175, 213, 310, 373, 528, 643, 862, 1044, 1403, 1699, 2199, 2676, 3426, 4131, 5256, 6295, 7884, 9479, 11722, 14047, 17296, 20623, 25142, 29942, 36299, 43081, 51950, 61439, 73668, 87040, 103748, 122149, 145155, 170487

Offset: 0

Gus Wiseman, Nov 08 2023

nonn

A multiset is non-knapsack if there exist two different submultisets with the same sum.

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (2111)  (321)    (3211)    (422)      (3321)
                 (2211)   (22111)   (431)      (4221)
                 (3111)   (31111)   (3221)     (4311)
                 (21111)  (211111)  (4211)     (5211)
                                    (22211)    (32211)
                                    (32111)    (33111)
                                    (41111)    (42111)
                                    (221111)   (222111)
                                    (311111)   (321111)
                                    (2111111)  (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

The complement is counted by A108917, strict A275972, ranks A299702.
These partitions have ranks A299729.
The strict case is A316402.
The binary version is A366753, ranks A366740.
A000041 counts integer partitions, strict A000009.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with subset-sum k, complement A046663.
A365661 counts strict partitions with subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
Cf. A002033, A006827, A122768, A126796, A238628, A365923, A365924, A367095.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#]]&]], {n,0,15}]

a(n) = A000041(n) - A108917(n).

A371793 Number of non-biquanimous subsets of {1..n} containing n.

Original entry on oeis.org

1, 2, 3, 6, 12, 22, 44, 84, 163, 314, 610, 1184, 2308, 4505, 8843, 17386, 34336, 67881, 134662, 267431, 532172, 1060048, 2113947, 4218325, 8423138, 16826162, 33623311, 67205646, 134351795, 268621562, 537124814, 1074092608, 2147953084, 4295613139, 8590784715, 17181035797, 34361248692, 68721546255, 137441586921, 274881519876, 549760320576, 1099517861045, 2199030848627, 4398057100987, 8796105652038, 17592203866158

Offset: 1

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(5) = 12 subsets:
  {1}  {2}    {3}    {4}      {5}
       {1,2}  {1,3}  {1,4}    {1,5}
              {2,3}  {2,4}    {2,5}
                     {3,4}    {3,5}
                     {1,2,4}  {4,5}
                     {2,3,4}  {1,2,5}
                              {1,3,5}
                              {2,4,5}
                              {3,4,5}
                              {1,2,3,5}
                              {1,3,4,5}
                              {1,2,3,4,5}

The complement is counted by A232466, differences of A371791.
This is the "bi-" version of A371790, differences of A371789.
First differences of A371792.
The complement is the "bi-" version of A371797, differences of A371796.
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, A365543, A365661, A365663, A366320, A365381, A367094, A371788.

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

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

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 4, 0, 2, 0, 1, 0, 5, 1, 0, 3, 1, 1, 0, 8, 0, 3, 0, 3, 0, 1, 0, 10, 2, 1, 2, 2, 3, 1, 1, 0, 16, 0, 5, 0, 3, 0, 5, 0, 1, 0, 20, 2, 2, 4, 2, 6, 0, 4, 1, 1, 0, 31, 0, 6, 0, 8, 0, 5, 0, 5, 0, 1, 0, 39, 4, 4, 4, 1, 6, 6, 3, 2, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   4  0  2  0  1  0
   5  1  0  3  1  1  0
   8  0  3  0  3  0  1  0
  10  2  1  2  2  3  1  1  0
  16  0  5  0  3  0  5  0  1  0
  20  2  2  4  2  6  0  4  1  1  0
  31  0  6  0  8  0  5  0  5  0  1  0
  39  4  4  4  1  6  6  3  2  6  1  1  0
  55  0 13  0  8  0 12  0  6  0  6  0  1  0
  71  5  8  7  3  5  3 16  3  6  0  6  1  1  0
Row n = 6 counts the following partitions:
  (321)     (411)  .  (51)   (33)  (6)  .
  (3111)              (42)
  (2211)              (222)
  (21111)
  (111111)

Row sums are A000041.
The rank statistic counted by this triangle is A325799.
The strict case is A365545, weighted row sums A365922.
The complement (positive subset-sum) is A365658.
Weighted row sums are A365918, for positive subset-sums A304792.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A000009, A006827, A122768, A364272, A365919, A365921.

Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]

A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 0, 5, 2, 0, 0, 5, 0, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Is column k = n - 7 given by A325695?

			Triangle begins:
  1
  1  0
  0  1  0
  1  0  1  0
  0  1  0  1  0
  0  0  2  0  1  0
  1  0  0  2  0  1  0
  1  0  0  0  3  0  1  0
  0  1  1  0  0  3  0  1  0
  0  0  3  0  0  0  4  0  1  0
  1  0  0  2  2  0  0  4  0  1  0
  1  0  0  0  5  0  0  0  5  0  1  0
  2  0  0  0  0  5  2  0  0  5  0  1  0
  2  0  1  0  0  0  8  0  0  0  6  0  1  0
  1  1  3  0  0  0  0  7  3  0  0  6  0  1  0
  2  0  4  0  1  0  0  0 12  0  0  0  7  0  1  0
  1  1  2  2  3  1  0  0  0 11  3  0  0  7  0  1  0
  2  0  3  0  7  0  1  0  0  0 16  0  0  0  8  0  1  0
  3  0  0  2  6  3  3  1  0  0  0 15  4  0  0  8  0  1  0
Row n = 12: counts the following partitions:
  (6,3,2,1)  .  .  .  .  (9,2,1)  (6,5,1)  .  .  (11,1)  .  (12)  .
  (5,4,2,1)              (8,3,1)  (6,4,2)        (10,2)
                         (7,4,1)                 (9,3)
                         (7,3,2)                 (8,4)
                         (5,4,3)                 (7,5)

Row sums are A000009, non-strict A000041.
The complement (positive subset-sums) is also A365545 with rows reversed.
Weighted row sums are A365922, non-strict A365918.
The non-strict version is A365923, complement A365658, rank stat A325799.
A046663 counts partitions without a subset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A284640, A304792, A364272, A365921.

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

A365922 Number of non-subset-sums of strict integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 11, 18, 25, 38, 51, 70, 93, 122, 159, 206, 263, 328, 420, 514, 645, 776, 967, 1154, 1413, 1686, 2042, 2414, 2890, 3394, 4062, 4732, 5598, 6494, 7652, 8836, 10329, 11884, 13833, 15830, 18376, 20936, 24131, 27476, 31547, 35780, 40966, 46292, 52737

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 11 ways, showing each strict partition and its non-subset-sums:
    (6): 1,2,3,4,5
   (51): 2,3,4
   (42): 1,3,5
  (321):

The complement (positive subset-sums) is A284640, non-strict A276024.
Weighted row sums of A365545, non-strict A365923.
Row sums of A365663, non-strict A046663.
The non-strict version is A365918.
The zero-full complement (subset-sums) is A365925, non-strict A304792.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
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.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A325799, A364272, A365658, A365919, A365921.

Table[Total[Length[Complement[Range[n], Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, May 15 2019

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  0  1
  0  1  2  2  0  1
  0  1  3  2  0  0  1
  0  1  3  4  2  0  0  1
  0  1  4  3  3  0  0  0  1
  0  1  4  7  2  2  0  0  0  1
  0  1  5  6  4  2  0  0  0  0  1
  0  1  5 10  6  4  2  0  0  0  0  1
  0  1  6  9  5  1  2  0  0  0  0  0  1
  0  1  6 14 10  5  2  2  0  0  0  0  0  1
  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1
  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1
Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)
        (75)   (543)   (5511)           (711111)
        (84)   (552)   (7221)
        (93)   (732)   (7311)
        (A2)   (741)   (9111)
        (B1)   (822)
               (831)
               (921)
               (A11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

Row sums are A000041.
Column k = 2 is A004526.
Column k = 3 is A325690.
Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

Table[Length[Select[IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15},{k,0,n}]

A365662 Number of ordered pairs of disjoint strict integer partitions of n.

Original entry on oeis.org

1, 0, 0, 2, 2, 6, 8, 14, 18, 32, 42, 66, 92, 136, 190, 280, 374, 532, 744, 1014, 1366, 1896, 2512, 3384, 4526, 6006, 7910, 10496, 13648, 17842, 23338, 30116, 38826, 50256, 64298, 82258, 105156, 133480, 169392, 214778, 270620, 340554, 428772, 536302, 670522

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.

			The a(0) = 1 through a(7) = 14 pairs:
  ()()  .  .  (21)(3)  (31)(4)  (32)(5)   (42)(6)   (43)(7)
              (3)(21)  (4)(31)  (41)(5)   (51)(6)   (52)(7)
                                (5)(32)   (6)(42)   (61)(7)
                                (5)(41)   (6)(51)   (7)(43)
                                (32)(41)  (321)(6)  (7)(52)
                                (41)(32)  (42)(51)  (7)(61)
                                          (51)(42)  (421)(7)
                                          (6)(321)  (43)(52)
                                                    (43)(61)
                                                    (52)(43)
                                                    (52)(61)
                                                    (61)(43)
                                                    (61)(52)
                                                    (7)(421)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

For subsets instead of partitions we have A000244, non-disjoint A000302.
If the partitions can have different sums we get A032302.
The non-strict version is A054440, non-disjoint A001255.
The unordered version is A108796, non-strict A260669.
A000041 counts integer partitions, strict A000009.
A000124 counts distinct possible sums of subsets of {1..n}.
A000712 counts distinct submultisets of partitions.
A002219 and A237258 count partitions of 2n including a partition of n.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 24 2025 *)

a(n) = 2*A108796(n) for n > 1.

a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - Ilya Gutkovskiy, Apr 24 2025

cp's OEIS Frontend

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A371782 Numbers with non-biquanimous prime signature.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A366754 Number of non-knapsack integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A371793 Number of non-biquanimous subsets of {1..n} containing n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A365922 Number of non-subset-sums of strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica