A365312 - OEIS

%I #16 Sep 14 2023 01:09:38

%S 0,0,0,1,1,3,2,6,4,8,7,16,6,24,17,24,20,46,22,62,31,63,57,106,35,122,

%T 90,137,88,212,74,262,134,267,206,345,121,476,294,484,232,698,242,837,

%U 389,763,571,1185,318,1327,634,1392,727,1927,640,2056,827,2233,1328

%N Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

%e The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).

%e The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).

%e The a(0) = 0 through a(11) = 16 strict partitions:

%e   .  .  .  (2)  (3)  (2)  (4)  (2)    (3)  (2)    (3)    (2)

%e                      (3)  (5)  (3)    (5)  (4)    (4)    (3)

%e                      (4)       (4)    (6)  (5)    (6)    (4)

%e                                (5)    (7)  (6)    (7)    (5)

%e                                (6)         (7)    (8)    (6)

%e                                (4,2)       (8)    (9)    (7)

%e                                            (4,2)  (6,3)  (8)

%e                                            (6,2)         (9)

%e                                                          (10)

%e                                                          (4,2)

%e                                                          (5,4)

%e                                                          (6,2)

%e                                                          (6,3)

%e                                                          (6,4)

%e                                                          (7,3)

%e                                                          (8,2)

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

%t Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]

%o (Python)

%o from math import isqrt

%o from sympy.utilities.iterables import partitions

%o def A365312(n):

%o     a = {tuple(sorted(set(p))) for p in partitions(n)}

%o     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 not any(set(d).issubset(set(b)) for d in a)) # _Chai Wah Wu_, Sep 13 2023

%Y The complement for positive coefficients is counted by A088314.

%Y For positive coefficients we have A088528.

%Y The complement is counted by A365311.

%Y For non-strict partitions we have A365378, complement A365379.

%Y The version for subsets is A365380, complement A365073.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A116861 and A364916 count linear combinations of strict partitions.

%Y A364350 counts combination-free strict partitions, non-strict A364915.

%Y A364839 counts combination-full strict partitions, non-strict A364913.

%Y Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

%K nonn

%O 0,6

%A _Gus Wiseman_, Sep 05 2023

%E a(26)-a(58) from _Chai Wah Wu_, Sep 13 2023