cp's OEIS Frontend

A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

%I A365312 #16 Sep 14 2023 01:09:38
%S A365312 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 A365312 90,137,88,212,74,262,134,267,206,345,121,476,294,484,232,698,242,837,
%U A365312 389,763,571,1185,318,1327,634,1392,727,1927,640,2056,827,2233,1328
%N A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.
%e A365312 The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
%e A365312 The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
%e A365312 The a(0) = 0 through a(11) = 16 strict partitions:
%e A365312   .  .  .  (2)  (3)  (2)  (4)  (2)    (3)  (2)    (3)    (2)
%e A365312                      (3)  (5)  (3)    (5)  (4)    (4)    (3)
%e A365312                      (4)       (4)    (6)  (5)    (6)    (4)
%e A365312                                (5)    (7)  (6)    (7)    (5)
%e A365312                                (6)         (7)    (8)    (6)
%e A365312                                (4,2)       (8)    (9)    (7)
%e A365312                                            (4,2)  (6,3)  (8)
%e A365312                                            (6,2)         (9)
%e A365312                                                          (10)
%e A365312                                                          (4,2)
%e A365312                                                          (5,4)
%e A365312                                                          (6,2)
%e A365312                                                          (6,3)
%e A365312                                                          (6,4)
%e A365312                                                          (7,3)
%e A365312                                                          (8,2)
%t A365312 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365312 Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]
%o A365312 (Python)
%o A365312 from math import isqrt
%o A365312 from sympy.utilities.iterables import partitions
%o A365312 def A365312(n):
%o A365312     a = {tuple(sorted(set(p))) for p in partitions(n)}
%o A365312     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 A365312 The complement for positive coefficients is counted by A088314.
%Y A365312 For positive coefficients we have A088528.
%Y A365312 The complement is counted by A365311.
%Y A365312 For non-strict partitions we have A365378, complement A365379.
%Y A365312 The version for subsets is A365380, complement A365073.
%Y A365312 A000041 counts integer partitions, strict A000009.
%Y A365312 A008284 counts partitions by length, strict A008289.
%Y A365312 A116861 and A364916 count linear combinations of strict partitions.
%Y A365312 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A365312 A364839 counts combination-full strict partitions, non-strict A364913.
%Y A365312 Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.
%K A365312 nonn
%O A365312 0,6
%A A365312 _Gus Wiseman_, Sep 05 2023
%E A365312 a(26)-a(58) from _Chai Wah Wu_, Sep 13 2023