A088528 -id:A088528 - cp's OEIS frontend

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

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}]

cp's OEIS Frontend

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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