A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

0, 1, 2, 3, 5, 6, 11, 12, 20, 24, 35, 38, 63, 63, 92, 112, 148, 160, 230, 244, 339, 383, 478, 533, 726, 781, 978, 1123, 1394, 1526, 1960, 2112, 2630, 2945, 3518, 3964, 4856, 5261, 6307, 7099, 8464, 9258, 11140, 12155, 14419, 16093, 18589, 20565, 24342, 26597, 30948

Offset

0,3

Links

Table of n, a(n) for n=0..50.

S. R. Finch, Monoids of natural numbers, March 17, 2009.

Example

The strict partition (6,3) cannot be linearly combined to obtain 10, so is not counted under a(10).
The strict partition (4,2) has 6 = 1*4 + 1*2 so is counted under a(6), but (4,2) cannot be linearly combined to obtain 7 so is not counted under a(7).
The a(1) = 1 through a(7) = 12 strict partitions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (2,1)  (4)    (2,1)  (3)      (2,1)
                   (2,1)  (3,1)  (6)      (3,1)
                   (3,1)  (3,2)  (2,1)    (3,2)
                          (4,1)  (3,1)    (4,1)
                                 (3,2)    (4,3)
                                 (4,1)    (5,1)
                                 (4,2)    (5,2)
                                 (5,1)    (6,1)
                                 (3,2,1)  (3,2,1)
                                          (4,2,1)

Mathematica

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

Prog

(Python)
from math import isqrt
from sympy.utilities.iterables import partitions
def A365311(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, m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

Crossrefs

For positive coefficients we have A088314.

The positive complement is counted by A088528.

The version for subsets is A365073.

The complement is counted by A365312.

For non-strict partitions we have 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. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

Sequence in context: A199366 A332275 A318689 * A083710 A127524 A117086

Adjacent sequences: A365308 A365309 A365310 * A365312 A365313 A365314

Keyword

nonn

Author

Gus Wiseman, Sep 04 2023

Extensions

a(26)-a(50) from Chai Wah Wu, Sep 13 2023

Status

approved