A122768 -id:A122768 - cp's OEIS frontend

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

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

A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 5, 3, 4, 2, 3, 7, 5, 6, 4, 3, 3, 11, 7, 9, 6, 6, 3, 4, 15, 11, 13, 10, 9, 6, 4, 4, 22, 15, 20, 13, 15, 9, 8, 4, 5, 30, 22, 27, 21, 21, 15, 12, 8, 5, 5, 42, 30, 39, 28, 30, 21, 20, 12, 10, 5, 6, 56, 42, 53, 41, 42, 33, 28, 20, 15, 10, 6, 6

Offset: 2

Gus Wiseman, Nov 17 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (3,2,1,1) has semi-sum 3 = 2+1, but no semi-sum 6, so y is counted under T(7,3) but not under T(7,6).
Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   2
   5   3   4   2   3
   7   5   6   4   3   3
  11   7   9   6   6   3   4
  15  11  13  10   9   6   4   4
  22  15  20  13  15   9   8   4   5
  30  22  27  21  21  15  12   8   5   5
  42  30  39  28  30  21  20  12  10   5   6
  56  42  53  41  42  33  28  20  15  10   6   6
  77  56  73  55  60  42  44  28  25  15  12   6   7
Row n = 7 counts the following partitions:
  (511)      (421)     (331)    (421)   (511)  (61)
  (4111)     (3211)    (322)    (4111)  (421)  (52)
  (3211)     (2221)    (3211)   (322)   (331)  (43)
  (31111)    (22111)   (31111)  (3211)
  (22111)    (211111)  (2221)
  (211111)             (22111)
  (1111111)

Column k = 0 is A000041.
Column n = k is A004526.
The complement for all submultisets is A046663, strict A365663.
For subsets instead of partitions we have A365541, non-binary A365381.
The non-binary version is A365543, strict A365661.
Row sums are A366738.
The strict case is A367405.
Cf. A122768, A108917, A299701, A304792, A364272, A364911, A365658.

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#, {2}],k]&]], {n,2,10}, {k,2,n}]

A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 2, 4, 3, 4, 4, 5, 3, 4, 5, 3, 3, 5, 4, 4, 5, 5, 5, 4, 4, 6, 4, 3, 6, 5, 6, 5, 7, 5, 7, 4, 5, 6, 5, 5, 7, 7, 8, 7, 8, 8, 7, 7, 5, 5, 7

Offset: 3

Gus Wiseman, Nov 18 2023

			Triangle begins:
  1
  0  1
  0  0  2
  1  1  1  2
  1  0  1  1  3
  1  1  1  1  2  3
  1  1  1  2  2  2  4
  2  2  3  2  3  2  3  4
  2  2  3  2  3  3  3  3  5
  3  2  4  3  4  4  5  3  4  5
  3  3  5  4  4  5  5  5  4  4  6
  4  3  6  5  6  5  7  5  7  4  5  6
  5  5  7  7  8  7  8  8  7  7  5  5  7
  6  5  9  8 10  7 10  9 10  7  9  5  6  7
  7  7 10 10 12 11 11 11 12 10  9  9  6  6  8
  9  7 13 11 15 12 13 13 15 13 13  9 11  6  7  8
Row n = 9 counts the following strict partitions:
  (6,2,1)  (5,3,1)  (4,3,2)  (5,3,1)  (6,2,1)  (6,2,1)  (8,1)
                             (4,3,2)  (4,3,2)  (5,3,1)  (7,2)
                                                        (6,3)
                                                        (5,4)
Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
  A21   931   841   751   652   751   841   931   A21  A21  C1
  7321  7321  832   742   643   7321  742   832   832  931  B2
  6421  5431  7321  6421  6421  652   7321  7321  742  841  A3
              6421  5431  5431  6421  643   643   652  751  94
              5431              5431  5431  6421            85
                                                            76

Column n = k is A004526.
Column k = 3 is A025148.
For subsets instead of partitions we have A365541, non-binary A365381.
The non-binary version is A365661, non-strict A365543.
The non-binary complement is A365663, non-strict A046663.
Row sums are A366741, non-strict A366738.
The non-strict version is A367404.
Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], k]&]], {n,3,10}, {k,3,n}]

A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Note that such a Heinz number is necessarily squarefree, as such a partition is necessarily strict.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
First differs from A301899 at a(43) = 70, because (4,3,1) is not knapsack but every submultiset has a different average.

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Cf. A000009, A056239, A122768, A237984, A275972, A299702, A301899, A316313, A316314, A316465.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Mean/@Union[Subsets[primeMS[#]]]&]

A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 28 2023

			The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  0  1
  0  1  0  1  0
  0  1  0  2  0  0
  0  1  0  2  0  0  1
  0  1  0  3  0  0  0  1
  0  1  0  3  0  0  1  1  0
  0  1  0  4  0  0  0  3  0  0
  0  1  0  4  0  0  2  2  0  0  1
  0  1  0  5  0  0  0  5  0  0  0  1
  0  1  0  5  0  0  2  5  0  0  0  0  2
  0  1  0  6  0  0  0  8  0  0  0  1  0  2
  0  1  0  6  0  0  3  7  0  0  0  0  3  1  1
  0  1  0  7  0  0  0 12  0  0  0  1  0  4  0  2
  0  1  0  7  0  0  3 11  0  0  0  1  3  2  2  1  1
  0  1  0  8  0  0  0 16  0  0  0  1  0  7  0  3  0  2
  0  1  0  8  0  0  4 15  0  0  0  1  3  3  6  2  0  0  3
  0  1  0  9  0  0  0 21  0  0  0  2  0  9  0  7  0  1  0  4
  0  1  0  9  0  0  4 20  0  0  1  0  4  8  5  5  0  0  2  0  5
Row n = 14 counts the following partitions (A..E = 10..14):
  (E)  .  (D1)  .  .  (761)  (B21)  .  .  .  .  (6521)  (8321)  (7421)
          (C2)        (752)  (A31)              (6431)
          (B3)        (743)  (941)              (5432)
          (A4)               (932)
          (95)               (851)
          (86)               (842)
                             (653)

Row sums are A000009.
Rightmost column n = k is A188431, non-strict A126796.
The one-based weighted row sums are A284640.
The corresponding rank statistic is A299701.
The non-strict version is A365658.
Central column n = 2k in the non-strict case is A365660.
Reverse-weighted row-sums are A365922, non-strict A276024.
A000041 counts integer partitions.
A000124 counts distinct sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A137719, A304792, A364916.

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

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

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

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 5, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 4, 3, 3, 4, 3, 3, 8, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 7, 6, 7, 7, 7, 4, 7, 7, 7, 6, 7, 15, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Feb 05 2014

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000

Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.

A301829 Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 15, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 29, 4, 4, 4, 37, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 64, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 54, 4, 17, 1, 12, 4, 17, 1, 92, 1, 4, 12, 12, 4, 17

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

			The a(12) = 12 submultisets ("<" means subset or equal):
(2)<(2*2*3), (3)<(2*2*3), (2*2)<(2*2*3), (2*3)<(2*2*3), (2*2*3)<(2*2*3),
(2)<(2*6), (6)<(2*6), (2*6)<(2*6),
(3)<(3*4), (4)<(3*4), (3*4)<(3*4),
(12)<(12).

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Rest[Divisors[n]]}],{n,100}]

a(n) = Sum_{d|n, d>1} f(d) * f(n/d) where f(n) = A001055(n) is the number of factorizations of n into factors greater than 1.

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[LCM@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A301935 Number of positive subset-sum trees whose composite a positive subset-sum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 10, 2, 3, 1, 21, 1, 3, 3, 58, 1, 21, 1, 21, 3, 3, 1, 164, 2, 3, 10, 21, 1, 34, 1, 373, 3, 3, 3, 218, 1, 3, 3, 161, 1, 7, 1, 5, 5, 3, 1, 1320, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 2558, 3, 7, 1, 5, 3, 6, 1, 7

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A positive subset-sum tree with root x is either the symbol x itself, or is obtained by first choosing a positive subset-sum x <= (y_1,...,y_k) with k > 1 and then choosing a positive subset-sum tree with root y_i for each i = 1...k. The composite of a positive subset-sum tree is the positive subset-sum x <= g where x is the root sum and g is the multiset of leaves. We write positive subset-sum trees in the form rootsum(branch,...,branch). For example, 4(1(1,3),2,2(1,1)) is a positive subset-sum tree with composite 4(1,1,1,2,3) and weight 8.

Gus Wiseman, The a(12) = 21 positive subset-sum trees.

Cf. A000108, A000712, A108917, A122768, A262671, A262673, A275972, A276024, A284640, A299701, A301854, A301855, A301856, A301934.

cp's OEIS Frontend

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

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A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

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A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

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A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

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A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

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A366753 Number of integer partitions of n without all different sums of two-element submultisets.

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A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

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A301829 Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

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A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

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