A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 7, 12, 12, 17, 20, 26, 29, 39, 43, 54, 62, 77, 88, 107, 122, 148, 168, 200, 229, 267, 308, 360, 407, 476, 536, 623, 710, 812, 917, 1050, 1190, 1349, 1530, 1733, 1944, 2206, 2483, 2794, 3138, 3524

Offset

0,7

Links

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

Example

For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
  .  .  (21)  (31)  (41)  (42)   (61)   (62)   (63)   (82)    (A1)    (84)
                          (51)   (421)  (71)   (81)   (91)    (542)   (93)
                          (321)         (431)  (432)  (532)   (632)   (A2)
                                        (521)  (531)  (541)   (641)   (B1)
                                               (621)  (631)   (731)   (642)
                                                      (721)   (821)   (651)
                                                      (4321)  (5321)  (732)
                                                                      (741)
                                                                      (831)
                                                                      (921)
                                                                      (5421)
                                                                      (6321)

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[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[combs[#[[k]], Delete[#, k]]!={}, {k, Length[#]}]&]], {n, 0, 15}]

Prog

(Python)
from sympy.utilities.iterables import partitions
def A364839(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):
        if max(p.values(), default=0)==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 23 2023

Crossrefs

For sums instead of combinations we have A364272, binary A364670.

The complement in strict partitions is A364350.

Non-strict versions are A364913 and the complement of A364915.

For subsets instead of partitions we have A364914, complement A326083.

The case of no all positive coefficients is A365006.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A116861 and A364916 count linear combinations of strict partitions.

Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.

Sequence in context: A334652 A163241 A234027 * A165279 A345866 A125060

Adjacent sequences: A364836 A364837 A364838 * A364840 A364841 A364842

Keyword

nonn

Author

Gus Wiseman, Aug 19 2023

Status

approved