A323092 -id:A323092 - cp's OEIS frontend

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325

Offset: 0

Gus Wiseman, Aug 15 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(16) = 6 through a(22) = 12 strict partitions:
  (16)     (17)     (18)     (19)     (20)      (21)      (22)
  (9,7)    (9,8)    (10,8)   (10,9)   (11,9)    (12,9)    (13,9)
  (10,6)   (10,7)   (11,7)   (11,8)   (12,8)    (13,8)    (14,8)
  (11,5)   (11,6)   (13,5)   (12,7)   (13,7)    (15,6)    (15,7)
  (13,3)   (12,5)   (14,4)   (13,6)   (14,6)    (16,5)    (16,6)
  (7,5,4)  (13,4)   (7,6,5)  (14,5)   (17,3)    (17,4)    (17,5)
           (14,3)   (8,7,3)  (15,4)   (8,7,5)   (19,2)    (18,4)
           (15,2)            (16,3)   (9,6,5)   (11,10)   (19,3)
           (7,6,4)           (17,2)   (9,7,4)   (8,7,6)   (12,10)
                             (8,6,5)  (11,5,4)  (9,7,5)   (9,7,6)
                             (9,6,4)            (10,7,4)  (9,8,5)
                                                (10,8,3)  (7,6,5,4)
                                                (11,6,4)
                                                (11,7,3)

For sums of subsets instead of combinations of partitions we have A151897.
For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The complement in strict partitions is A364839, non-strict A364913.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A116861 and A364916 count linear combinations of strict partitions.
A323092 (ranks A320340) and A120641 count double-free partitions.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combs[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364350(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

More terms and offset corrected by Martin Fuller, Sep 11 2023

A364272 Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 11, 10, 17, 16, 26, 25, 39, 39, 54, 60, 82, 84, 116, 126, 160, 177, 222, 242, 302, 337, 402, 453, 542, 601, 722, 803, 936, 1057, 1234, 1373, 1601, 1793, 2056, 2312, 2658, 2950, 3395, 3789, 4281, 4814, 5452, 6048

Offset: 0

Gus Wiseman, Aug 01 2023

nonn

First differs from A316402 at a(16) = 11 due to (7,5,3,1).

			The a(6) = 1 through a(16) = 11 partitions (A=10):
  (321) . (431) . (532)  (5321) (642)  (5431) (743)  (6432)  (853)
                  (541)         (651)  (6421) (752)  (6531)  (862)
                  (4321)        (5421) (7321) (761)  (7431)  (871)
                                (6321)        (5432) (7521)  (6532)
                                              (6431) (9321)  (6541)
                                              (6521) (54321) (7432)
                                              (7421)         (7621)
                                              (8321)         (8431)
                                                             (8521)
                                                             (A321)
                                                             (64321)

The non-strict complement is A237667, ranks A364531.
The non-strict version is A237668, ranks A364532.
The complement in strict partitions is A364349, binary A364533.
The linear combination-free version is A364350.
For subsets of {1..n} we have A364534, complement A151897.
The binary version is A364670, allowing re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972, ranks A299702.
A236912 counts binary sum-free partitions, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363225, A364346.

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

A237113 Number of partitions of n such that some part is a sum of two other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326

Offset: 0

Clark Kimberling, Feb 04 2014

nonn

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 3.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(0) = 0 through a(9) = 10 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (21111)  (211111)  (3221)     (4311)
                                                (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (42111)
                                                (221111)   (222111)
                                                (2111111)  (321111)
                                                           (2211111)
                                                           (21111111)
(End)

The complement for subsets is A085489, with re-usable parts A007865.
For subsets of {1..n} we have A088809, with re-usable parts A093971.
The complement is counted by A236912, ranks A364461.
The non-binary complement is A237667, ranks A364531.
The non-binary version is A237668, ranks A364532.
With re-usable parts we have A363225, ranks A364348.
The complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364462.
The strict case is A364670, with re-usable parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A151897, A237984, A325862, A326083, A363260.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,30}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A236912(n).

a(0)=0 prepended by Alois P. Heinz, Sep 17 2023

A236912 Number of partitions of n such that no part is a sum of two other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 12, 14, 20, 25, 34, 40, 54, 64, 85, 98, 127, 149, 189, 219, 277, 316, 395, 456, 557, 638, 778, 889, 1070, 1226, 1461, 1667, 1978, 2250, 2645, 3019, 3521, 3997, 4652, 5267, 6093, 6909, 7943, 8982, 10291, 11609, 13251, 14947, 16984, 19104

Offset: 0

Clark Kimberling, Feb 01 2014

nonn

These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - Gus Wiseman, Aug 09 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 11 - 3 = 8.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (3111)    (421)      (521)
                                     (111111)  (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)
(End)

For subsets of {1..n} we have A085489, complement A088809.
The complement is counted by A237113, ranks A364462.
The non-binary version is A237667, ranks A364531.
The non-binary complement is A237668, ranks A364532.
The version with re-usable parts is A364345, ranks A364347.
The (strict) version for linear combinations of parts is A364350.
These partitions have ranks A364461.
The strict case is A364533, non-binary A364349.
The strict complement is A364670, with re-usable parts A363226.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A237113(n).

a(0)=1 prepended by Alois P. Heinz, Sep 17 2023

A237667 Number of partitions of n such that no part is a sum of two or more other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 61, 61, 87, 85, 120, 117, 160, 156, 224, 216, 288, 277, 380, 363, 483, 474, 622, 610, 783, 755, 994, 986, 1235, 1191, 1549, 1483, 1876, 1865, 2306, 2279, 2806, 2732, 3406, 3413, 4091, 4013, 4991, 4895, 5872

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

From Gus Wiseman, Aug 09 2023: (Start)
Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).
These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.
(End)

			For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
From _Gus Wiseman_, Aug 09 2023: (Start)
The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

For subsets of {1..n} we have A151897, binary A085489.
The binary version is A236912, ranks A364461.
The binary complement is A237113, ranks A364462.
The complement is counted by A237668, ranks A364532.
The binary version with re-usable parts is A364345, strict A364346.
The strict case is A364349, binary A364533.
These partitions have ranks A364531.
The complement for subsets is A364534, binary A088809.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A179009, A325862, A326083, A363225, A363226, A364348, A364350, A364670.

Map[Count[Map[MemberQ[#,Apply[Alternatives,Map[Apply[Plus,#]&, DeleteDuplicates[DeleteCases[Subsets[#],?(Length[#]<2&)]]]]]&, IntegerPartitions[#]],False]&,Range[20]]  (* _Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(21)-a(53) from Giovanni Resta, Feb 22 2014

A237668 Number of partitions of n such that some part is a sum of two or more other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

			a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(0) = 0 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)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

Cf. A179009.
The binary complement is A236912, ranks A364461.
The binary version is A237113, ranks A364462.
The complement is counted by A237667, ranks A364531.
The binary version with re-usable parts is A363225, ranks A364348.
The strict case is A364272.
The binary complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts distinct subset-sums of prime indices.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* Gus Wiseman, Aug 12 2023 *)

a(21)-a(47) from Giovanni Resta, Feb 22 2014

A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 27, 68, 156, 357, 775, 1667, 3505, 7303, 15019, 30759, 62489, 126619, 255542, 514721, 1034425, 2076924, 4164650, 8346306, 16715847, 33467324, 66982798, 134040148, 268179417, 536510608, 1073226084, 2146759579, 4293930436, 8588485846, 17177799658

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(0) = 0 through a(5) = 10 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,3,4}    {1,3,4}
                    {1,2,3,4}  {1,4,5}
                               {2,3,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Andrew Howroyd, Table of n, a(n) for n = 0..85

The binary version is A088809, complement A085489.
The complement is counted by A151897.
The complement for partitions is A237667, ranks A364531.
For partitions we have A237668, ranks A364532.
For strict partitions we have A364272, complement A364349.
A108917 counts knapsack partitions, strict A275972.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
Cf. A007865, A093971, A323092, A325862, A326083, A363225, A364345, A364346, A364348, A364350, A364533, A364670.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,10}]

a(n) = 2^n - A151897(n). - Andrew Howroyd, Jan 27 2024

a(16)-a(25) from Chai Wah Wu, Nov 14 2023

a(26) onwards (using A151897) added by Andrew Howroyd, Jan 27 2024

A363225 Number of integer partitions of n containing three parts (a,b,c) (repeats allowed) such that a + b = c. A variation of sum-full partitions.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 29, 43, 58, 81, 109, 148, 195, 263, 339, 445, 574, 744, 942, 1209, 1515, 1923, 2399, 3005, 3721, 4629, 5693, 7024, 8589, 10530, 12804, 15596, 18876, 22870, 27538, 33204, 39816, 47766, 57061, 68161, 81099, 96510, 114434, 135634

Offset: 0

Gus Wiseman, Jul 19 2023

nonn

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (432)
                       (2211)   (3211)    (521)      (621)
                       (21111)  (22111)   (3221)     (3321)
                                (211111)  (4211)     (4221)
                                          (22211)    (4311)
                                          (32111)    (5211)
                                          (221111)   (22221)
                                          (2111111)  (32211)
                                                     (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

For subsets of {1..n} we have A093971, A088809 without re-using parts.
The complement for subsets is A007865, A085489 without re-using parts.
Without re-using parts we have A237113, complement A236912.
For sums of any length > 1 (without re-usable parts) we have A237668, complement A237667.
The strict case is A363226.
The complement is counted by A364345, strict A364346.
These partitions have ranks A364348, complement A364347.
The strict linear combination-free version is A364350.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A108917, A237984, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,15}]

from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A363225(n): return sum(1 for p in partitions(n) if any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 21 2023

a(31)-a(48) from Chai Wah Wu, Sep 21 2023

A320340 Heinz numbers of double-free integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 07 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is double-free if no part is twice any other part.
Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 2, 0, 1, 0, 4, 1, 1, 3, 1, 1, 0, 5, 2, 2, 2, 3, 0, 1, 0, 6, 2, 4, 2, 3, 3, 1, 1, 0, 8, 3, 4, 4, 3, 2, 5, 0, 1, 0, 10, 3, 5, 4, 7, 4, 3, 4, 1, 1, 0, 12, 5, 6, 6, 7, 7, 4, 3, 5, 0, 1, 0, 15, 5, 9, 7, 8, 6, 12, 3, 4, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Aug 17 2023

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.

			Array begins:
  1  1  1  2  2  3  4   5   6   8   10   12  15   18   22   27
  0  1  0  1  1  1  2   2   3   3   5    5   7    8    10   12
  0  1  1  2  1  2  4   4   5   6   9    10  13   15   19   23
  0  1  0  3  2  2  4   4   6   7   11   11  15   17   22   27
  0  1  1  3  3  3  7   7   8   10  16   17  23   27   33   42
  0  1  0  3  2  4  7   6   9   9   17   17  23   26   33   43
  0  1  1  5  3  4  12  10  13  16  26   27  36   42   52   68
  0  1  0  4  3  3  10  11  13  13  27   25  35   40   51   67
  0  1  1  5  4  5  15  13  19  20  36   37  51   58   72   97
  0  1  0  6  4  5  14  13  18  23  42   39  54   61   78   105
  0  1  1  6  4  6  20  17  23  25  54   50  69   80   98   138
  0  1  0  6  4  5  19  16  23  24  54   55  71   80   103  144
  0  1  1  8  6  7  27  23  30  35  72   70  103  113  139  199
  0  1  0  7  5  6  24  21  29  31  75   68  95   115  139  201
  0  1  1  8  5  7  31  27  36  39  90   86  122  137  178  255
  0  1  0  9  6  8  31  27  38  42  100  93  129  148  187  289
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   3  1  2  0  1  0
   4  1  1  3  1  1  0
   5  2  2  2  3  0  1  0
   6  2  4  2  3  3  1  1  0
   8  3  4  4  3  2  5  0  1  0
  10  3  5  4  7  4  3  4  1  1  0
  12  5  6  6  7  7  4  3  5  0  1  0
  15  5  9  7  8  6 12  3  4  6  1  1  0
  18  7 10 11 10  9 10 10  5  4  6  0  1  0
  22  8 13 11 16  9 13 11 15  5  4  6  1  1  0
  27 10 15 15 17 17 16 13 13 14  6  4  8  0  1  0

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Same as A116861 with offset 0 and rows reversed, non-strict version A364912.
Row n = 0 is A000009.
Row n = 1 is A096765.
Row n = 2 is A365005.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A000035.
Column k = 3 is A137719.
The main diagonal is A364910.
Left half has row sums A365002.
For not just strict partitions we have A365004, diagonal A364907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A066328 adds up distinct prime indices.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
t[n_,k_]:=Length[Join@@Table[combs[n,ptn],{ptn,Select[IntegerPartitions[k],UnsameQ@@#&]}]];
Table[t[k,n-k],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

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A364272 Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.

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A237113 Number of partitions of n such that some part is a sum of two other parts.

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A236912 Number of partitions of n such that no part is a sum of two other parts.

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A237667 Number of partitions of n such that no part is a sum of two or more other parts.

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A237668 Number of partitions of n such that some part is a sum of two or more other parts.

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A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

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A363225 Number of integer partitions of n containing three parts (a,b,c) (repeats allowed) such that a + b = c. A variation of sum-full partitions.

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