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A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

%I A364350 #19 Sep 24 2023 04:16:25
%S A364350 1,1,1,1,1,2,1,3,2,3,3,5,3,6,5,7,6,9,7,11,10,14,12,16,15,20,17,24,22,
%T A364350 27,29,32,30,41,36,49,45,50,52,65,63,70,77,80,83,104,98,107,116,126,
%U A364350 134,152,148,162,180,196,195,227,227,238,272,271,293,333,325
%N A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.
%C A364350 A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
%e A364350 The a(16) = 6 through a(22) = 12 strict partitions:
%e A364350   (16)     (17)     (18)     (19)     (20)      (21)      (22)
%e A364350   (9,7)    (9,8)    (10,8)   (10,9)   (11,9)    (12,9)    (13,9)
%e A364350   (10,6)   (10,7)   (11,7)   (11,8)   (12,8)    (13,8)    (14,8)
%e A364350   (11,5)   (11,6)   (13,5)   (12,7)   (13,7)    (15,6)    (15,7)
%e A364350   (13,3)   (12,5)   (14,4)   (13,6)   (14,6)    (16,5)    (16,6)
%e A364350   (7,5,4)  (13,4)   (7,6,5)  (14,5)   (17,3)    (17,4)    (17,5)
%e A364350            (14,3)   (8,7,3)  (15,4)   (8,7,5)   (19,2)    (18,4)
%e A364350            (15,2)            (16,3)   (9,6,5)   (11,10)   (19,3)
%e A364350            (7,6,4)           (17,2)   (9,7,4)   (8,7,6)   (12,10)
%e A364350                              (8,6,5)  (11,5,4)  (9,7,5)   (9,7,6)
%e A364350                              (9,6,4)            (10,7,4)  (9,8,5)
%e A364350                                                 (10,8,3)  (7,6,5,4)
%e A364350                                                 (11,6,4)
%e A364350                                                 (11,7,3)
%t A364350 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A364350 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combs[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,15}]
%o A364350 (Python)
%o A364350 from sympy.utilities.iterables import partitions
%o A364350 def A364350(n):
%o A364350     if n <= 1: return 1
%o A364350     alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
%o A364350     for p in partitions(n,k=n-1):
%o A364350         if max(p.values(),default=0)==1:
%o A364350             s = set(p)
%o A364350             if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
%o A364350                 c += 1
%o A364350     return c # _Chai Wah Wu_, Sep 23 2023
%Y A364350 For sums of subsets instead of combinations of partitions we have A151897.
%Y A364350 For sums instead of combinations we have A237667, binary A236912.
%Y A364350 For subsets instead of partitions we have A326083, complement A364914.
%Y A364350 The complement in strict partitions is A364839, non-strict A364913.
%Y A364350 A more strict variation is A364915.
%Y A364350 The case of all positive coefficients is A365006.
%Y A364350 A000041 counts integer partitions, strict A000009.
%Y A364350 A008284 counts partitions by length, strict A008289.
%Y A364350 A108917 counts knapsack partitions, ranks A299702.
%Y A364350 A116861 and A364916 count linear combinations of strict partitions.
%Y A364350 A323092 (ranks A320340) and A120641 count double-free partitions.
%Y A364350 A364912 counts linear combinations of partitions of k.
%Y A364350 Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.
%K A364350 nonn
%O A364350 0,6
%A A364350 _Gus Wiseman_, Aug 15 2023
%E A364350 More terms and offset corrected by _Martin Fuller_, Sep 11 2023