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

1,5

Links

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

Example

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)

Mathematica

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}]

Prog

(Python)
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

Crossrefs

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.

Sequence in context: A200361 A222471 A180858 * A263193 A094885 A240199

Adjacent sequences: A365320 A365321 A365322 * A365324 A365325 A365326

Keyword

nonn

Author

Gus Wiseman, Sep 12 2023

Extensions

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

Status

approved