A364915 -id:A364915 - cp's OEIS frontend

A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

1, 1, 4, 13, 50, 179, 696, 2619, 10119, 38867, 150407, 582065, 2260367, 8786919, 34225256, 133471650, 521216494, 2037608462, 7974105052, 31235316275, 122457794193, 480473181271, 1886555402750, 7412471695859, 29142658077266, 114643347181003, 451237737215201

Offset: 0

Gus Wiseman, Aug 18 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(0) = 1 through a(3) = 13 ways:
  0  1*1  1*2      1*3
          0*1+2*1  0*2+3*1
          1*1+1*1  1*2+1*1
          2*1+0*1  0*1+0*1+3*1
                   0*1+1*1+2*1
                   0*1+2*1+1*1
                   0*1+3*1+0*1
                   1*1+0*1+2*1
                   1*1+1*1+1*1
                   1*1+2*1+0*1
                   2*1+0*1+1*1
                   2*1+1*1+0*1
                   3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A000041.
A finer version is A364906.
The version for compositions is A364908, strict A364909.
Using just strict partitions we get A364910, main diagonal of A364916.
Main diagonal of A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A116861, A364350, A364839, A364911, A364912, A364913, A364914, A364915, A365002, A365003.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..27);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[n]}]],{n,0,5}]

a(n) = Sum_{m:A056239(m)=n} A364906(m).
a(n) = A364912(2n,n).
a(n) = A365004(n,n).

a(9)-a(26) from Alois P. Heinz, Jan 28 2024

A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 2, 9, 5, 13, 10, 28, 7, 45, 25, 51, 32, 101, 31, 148, 50, 166, 106, 291, 47, 374, 176, 450, 179, 721, 121, 963, 285, 1080, 474, 1534, 200, 2140, 712, 2407, 599, 3539, 481, 4546, 1014, 4885

Offset: 0

Gus Wiseman, Sep 04 2023

			The partition (5,2,2) has distinct parts {2,5} and has 11 = 3*2 + 1*5, so is not counted under a(11).
The partition (4,2,2) cannot be linearly combined to obtain 9, so is counted under a(9).
The partition (4,2,2) has distinct parts {2,4} and has 10 = 5*2 + 0*4, so is not counted under a(10).
The a(3) = 1 through a(10) = 10 partitions:
  (2)  (3)  (2)   (4)  (2)    (3)   (2)     (3)
            (3)   (5)  (3)    (5)   (4)     (4)
            (4)        (4)    (6)   (5)     (6)
            (22)       (5)    (7)   (6)     (7)
                       (6)    (33)  (7)     (8)
                       (22)         (8)     (9)
                       (33)         (22)    (33)
                       (42)         (42)    (44)
                       (222)        (44)    (63)
                                    (62)    (333)
                                    (222)
                                    (422)
                                    (2222)

The complement for subsets is A365073, positive coefficients A088314.
For strict partitions we have A365312, positive coefficients A088528.
For positive coefficients we have A365323.
The complement is counted by A365379.
The version for subsets is A365380, positive coefficients A365322.
The relatively prime case is A365382.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combs[n,Union[#]]=={}&]],{n,0,10}]

from sympy.utilities.iterables import partitions
def A365378(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n) for b in partitions(m) if not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(21)-a(45) from Chai Wah Wu, Sep 13 2023

A365379 Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

			The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
  (1)  (1)   (1)    (1)     (1)
       (2)   (3)    (2)     (5)
       (11)  (11)   (4)     (11)
             (21)   (11)    (21)
             (111)  (21)    (31)
                    (22)    (32)
                    (31)    (41)
                    (111)   (111)
                    (211)   (211)
                    (1111)  (221)
                            (311)
                            (1111)
                            (2111)
                            (11111)

For subsets with positive coefficients we have A088314, complement A088528.
The case of strict partitions with positive coefficients is also A088314.
The version for subsets is A365073, complement A365380.
The case of strict partitions is A365311, complement A365312.
The complement is counted by A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]

from sympy.utilities.iterables import partitions
def A365379(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(21)-a(43) from Chai Wah Wu, Sep 13 2023

A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 8, 4, 11, 9, 16, 14, 25, 20, 37, 31, 49, 47, 73, 64, 101, 96, 135, 133, 190, 181, 256, 253, 336, 342, 453, 452, 596, 609, 771, 803, 1014, 1041, 1309, 1362, 1674, 1760, 2151, 2249, 2736, 2884, 3449, 3661, 4366, 4615, 5486, 5825

Offset: 0

Gus Wiseman, Aug 31 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

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

Chai Wah Wu, Table of n, a(n) for n = 0..109

The nonnegative version for subsets appears to be A124506.
For sums instead of combinations we have A364349, binary A364533.
The nonnegative version is A364350, complement A364839.
For subsets instead of partitions we have A365044, complement A365043.
The non-strict version is A365072, nonnegative A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364912 counts linear combinations of partitions of k.
Cf. A151897, A237113, A236912, A237667, A275972, A363226, A364272, A364913, A365004, A365068.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combp[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,30}]

from sympy.utilities.iterables import partitions
def A365006(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        if max(p.values()) == 1:
            s = set(p)
            for q in s:
                if tuple(sorted(s-{q})) in alist[q]:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(56) from Chai Wah Wu, Sep 20 2023

A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 10, 16, 23, 34, 44, 67, 85, 119, 157, 210, 268, 360, 453, 592, 748, 956, 1195, 1520, 1883, 2365, 2920, 3628, 4451, 5494, 6702, 8211, 9976, 12147, 14666, 17776, 21389, 25774, 30887, 37035, 44224, 52819, 62836, 74753, 88614, 105062, 124160

Offset: 0

Gus Wiseman, Aug 27 2023

nonn

These may be called "non-binary nonnegative combination-full" partitions.

Does not necessarily include all non-strict partitions (A047967).

			The partition (5,4,3,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(15).
The partition (6,4,3,2) has 6 = 1*2 + 1*4, so is counted under a(15). The combinations 6 = 2*3 = 3*2 and 4 = 2*2 can also be used.
The a(3) = 1 through a(8) = 16 partitions:
  (21)  (31)   (41)    (42)     (61)      (62)
        (211)  (221)   (51)     (331)     (71)
               (311)   (321)    (421)     (422)
               (2111)  (411)    (511)     (431)
                       (2211)   (2221)    (521)
                       (3111)   (3211)    (611)
                       (21111)  (4111)    (3221)
                                (22111)   (3311)
                                (31111)   (4211)
                                (211111)  (5111)
                                          (22211)
                                          (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

The complement for sums instead of combinations is A237667, binary A236912.
For sums instead of combinations we have A237668, binary A237113.
The strict case is A364839, complement A364350.
Allowing equal parts in the combination gives A364913.
For subsets instead of partitions we have A364914, complement A326083.
The complement is A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A323092 counts double-free partitions, ranks A320340.
A364912 counts linear combinations of partitions of k.
Cf. A108917, A151897, A364272, A364910, A364911, A365006.

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], Function[ptn,Or@@Table[combs[ptn[[k]], DeleteCases[ptn,ptn[[k]]]]!={}, {k,Length[ptn]}]]]],{n,0,5}]

from sympy.utilities.iterables import partitions
def A365068(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        s = set(p)
        if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(47) from Chai Wah Wu, Sep 20 2023

A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 2, 4, 12, 8, 20, 11, 14, 26, 43, 19, 38, 53, 51, 48, 101, 48, 124, 96, 121, 159, 134, 103, 241, 261, 244, 175, 401, 229, 488, 358, 328

Offset: 0

Gus Wiseman, Sep 08 2023

			The a(11) = 2 through a(18) = 8 partitions:
  (5,4)  .  (6,5)  (6,5)   (7,6)  (7,5)   (7,4)     (7,5)
  (7,3)     (7,4)  (8,5)   (9,4)  (7,6)   (7,6)     (8,7)
            (7,5)  (9,4)          (9,5)   (8,5)     (10,7)
            (8,3)  (10,3)         (11,3)  (8,7)     (11,4)
                                          (9,5)     (11,5)
                                          (9,7)     (12,5)
                                          (10,3)    (13,4)
                                          (11,4)    (7,5,5)
                                          (11,5)
                                          (13,3)
                                          (7,4,4)
                                          (10,3,3)

Relatively prime partitions are counted by A000837, ranks A289509.
This is the relatively prime case of A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A007359, A289508, A364345, A365073, A365312, A365379, A365380, A365383.

combsu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],GCD@@#==1&&combsu[n,#]=={}&]],{n,0,20}]

from math import gcd
from sympy.utilities.iterables import partitions
def A365382(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n) for b in partitions(m) if gcd(*b.keys()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(21)-a(45) from Chai Wah Wu, Sep 13 2023

A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 17, 15, 31, 34, 53, 65, 109, 117, 196, 224, 328, 405, 586, 673, 968, 1163, 1555, 1889, 2531, 2986, 3969, 4744, 6073, 7333, 9317, 11053, 14011, 16710, 20702, 24714, 30549, 36127, 44413, 52561, 63786, 75583, 91377, 107436, 129463

Offset: 0

Gus Wiseman, Aug 31 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (52)       (53)
                                     (111111)  (322)      (332)
                                               (1111111)  (2222)
                                                          (11111111)
The a(11) = 17 partitions:
  (11)  (9,2)  (7,2,2)  (5,3,2,1)  (4,3,2,1,1)  (1,1,1,1,1,1,1,1,1,1,1)
        (8,3)  (6,3,2)  (5,2,2,2)  (3,2,2,2,2)
        (7,4)  (5,4,2)  (4,3,2,2)
        (6,5)  (5,3,3)  (3,3,3,2)
               (4,4,3)

The nonnegative version is A364915, strict A364350.
The strict case is A365006.
For subsets instead of partitions we have A365044, complement A365043.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A237667 counts sum-free partitions, binary A236912.
A364912 counts positive linear combinations of partitions.
A365068 counts combination-full partitions, strict A364839.
Cf. A085489, A108917, A151897, A325862, A364272, A364910, A364911, A364913.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Union/@IntegerPartitions[n], Function[ptn,!Or@@Table[combp[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A365072(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        s = set(p)
        for q in s:
            if tuple(sorted(s-{q})) in alist[q]:
                break
        else:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(49) from Chai Wah Wu, Sep 20 2023

A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 9, 7, 15, 16, 29, 23, 47, 43, 74, 65, 114, 100, 174, 153, 257, 228, 368, 312, 530, 454, 736, 645, 1025, 902, 1402, 1184, 1909, 1626, 2618, 2184, 3412, 2895, 4551, 3887, 5966, 5055, 7796, 6509, 10244, 8462, 13060, 10881, 16834, 14021, 21471

Offset: 1

Gus Wiseman, Sep 12 2023

nonn

			The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
The a(3) = 1 through a(10) = 16 partitions:
  (2)  (3)  (2)    (4)    (2)      (3)    (2)        (3)
            (3)    (5)    (3)      (5)    (4)        (4)
            (4)    (3,2)  (4)      (6)    (5)        (6)
            (2,2)         (5)      (7)    (6)        (7)
                          (6)      (3,3)  (7)        (8)
                          (2,2)    (4,3)  (8)        (9)
                          (3,3)    (5,2)  (2,2)      (3,3)
                          (4,2)           (4,2)      (4,4)
                          (2,2,2)         (4,3)      (5,2)
                                          (4,4)      (5,3)
                                          (5,3)      (5,4)
                                          (6,2)      (6,3)
                                          (2,2,2)    (7,2)
                                          (4,2,2)    (3,3,3)
                                          (2,2,2,2)  (4,3,2)
                                                     (5,2,2)

Chai Wah Wu, Table of n, a(n) for n = 1..95

Complement for subsets: A088314 or A365042, nonnegative A365073 or A365542.
For strict partitions we have A088528, nonnegative coefficients A365312.
For length-2 subsets we have A365321 (we use n instead of n-1).
For subsets we have A365322 or A365045, nonnegative coefficients A365380.
For nonnegative coefficients we have A365378, complement A365379.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]

from sympy.utilities.iterables import partitions
def A365323(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # Chai Wah Wu, Sep 12 2023

a(21)-a(51) from Chai Wah Wu, Sep 12 2023

A365383 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53

Offset: 0

Gus Wiseman, Sep 08 2023

Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
  22, 15, 19, 17, 20, 17, 20, 16,
  22, 17, 20, 17, 21, 17, 20, 17,
  22, 17, 20, 17, 21, 17, 20, 17, ...

			Triangle begins:
   1
   2   1
   3   2   2
   5   3   4   3
   7   5   6   6   6
  11   7   9   8   9   7
  15  11  13  13  14  13  14
  22  15  19  17  20  17  20  16
  30  22  26  26  27  26  28  26  27
  42  30  37  34  39  33  40  34  39  34
  56  42  50  49  52  50  54  51  54  53  53
  77  56  68  64  71  63  73  63  71  65  70  62
 101  77  91  89  95  90  97  93  97  97  98  94  99
 135 101 122 115 127 115 130 114 131 119 130 117 132 116
 176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
  (6)       (51)      (51)      (51)      (51)      (51)
  (51)      (411)     (42)      (411)     (42)      (411)
  (42)      (321)     (411)     (33)      (411)     (321)
  (411)     (3111)    (321)     (321)     (321)     (3111)
  (33)      (2211)    (3111)    (3111)    (3111)    (2211)
  (321)     (21111)   (222)     (2211)    (222)     (21111)
  (3111)    (111111)  (2211)    (21111)   (2211)    (111111)
  (222)               (21111)   (111111)  (21111)
  (2211)              (111111)            (111111)
  (21111)
  (111111)

Column k = 0 is A000041, strict A000009.
The version for subsets is A365381, main diagonal A365376.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A088314, A088528, A237668, A363225, A364345, A365073, A365311, A365320, A365378, A365382.

combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]

A365005 Number of ways to write 2 as a nonnegative linear combination of a strict integer partition of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 4, 4, 5, 6, 9, 10, 13, 15, 19, 23, 28, 33, 40, 47, 56, 67, 78, 92, 108, 126, 146, 171, 198, 229, 264, 305, 350, 403, 460, 527, 603, 687, 781, 889, 1009, 1144, 1295, 1464, 1653, 1866, 2101, 2364, 2659, 2984, 3347, 3752, 4200, 4696, 5248, 5858

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

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).

			The a(6) = 4 ways:
  0*5 + 2*1
  0*4 + 1*2
  0*3 + 0*2 + 2*1
  0*3 + 1*2 + 0*1

For 1 instead of 2 we have A096765.
Column k = n - 2 of A116861.
Row n = 2 of A364916.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A137719, A237113, A323092, A365004, A364272, A364907, A364910, A364914, A364915, A365002.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[2,ptn], {ptn,Select[IntegerPartitions[n], UnsameQ@@#&]}]],{n,0,30}]

cp's OEIS Frontend

A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

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A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.

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A365379 Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

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A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

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A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.

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A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.

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A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

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A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.

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