cp's OEIS Frontend

A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

%I A367218 #17 Dec 30 2023 17:00:35
%S A367218 1,1,1,2,4,6,8,13,18,26,35,50,66,92,119,160,208,275,350,457,579,742,
%T A367218 933,1185,1476,1859,2300,2868,3531,4371,5343,6575,8003,9776,11842,
%U A367218 14394,17351,20987,25191,30315,36257,43448,51753,61776,73342,87192,103184,122253,144211
%N A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.
%C A367218 The Heinz numbers of these partitions are given by A367226.
%e A367218 The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
%e A367218 The a(1) = 1 through a(7) = 13 partitions:
%e A367218   (1)  (11)  (21)   (22)    (32)     (42)      (52)
%e A367218              (111)  (31)    (41)     (51)      (61)
%e A367218                     (211)   (221)    (321)     (322)
%e A367218                     (1111)  (311)    (411)     (331)
%e A367218                             (2111)   (2211)    (421)
%e A367218                             (11111)  (3111)    (511)
%e A367218                                      (21111)   (2221)
%e A367218                                      (111111)  (3211)
%e A367218                                                (4111)
%e A367218                                                (22111)
%e A367218                                                (31111)
%e A367218                                                (211111)
%e A367218                                                (1111111)
%t A367218 combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t A367218 Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]!={}&]], {n,0,15}]
%Y A367218 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
%Y A367218                sum-full   sum-free   comb-full  comb-free
%Y A367218               -------------------------------------------
%Y A367218   partitions:  A367212    A367213    A367218*   A367219
%Y A367218   strict:      A367214    A367215    A367220    A367221
%Y A367218   subsets:     A367216    A367217    A367222    A367223
%Y A367218   ranks:       A367224    A367225    A367226    A367227
%Y A367218 A000041 counts integer partitions, strict A000009.
%Y A367218 A002865 counts partitions whose length is a part, complement A229816.
%Y A367218 A008284 counts partitions by length, strict A008289.
%Y A367218 A240855 counts strict partitions whose length is a part, complement A240861.
%Y A367218 A365046 counts combination-full subsets, differences of A364914.
%Y A367218 Cf. A088314, A363225, A364350, A365073, A365311, A365918.
%K A367218 nonn
%O A367218 0,4
%A A367218 _Gus Wiseman_, Nov 14 2023
%E A367218 a(31)-a(48) from _Chai Wah Wu_, Nov 15 2023