cp's OEIS Frontend

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

%I A367220 #6 Nov 15 2023 08:23:08
%S A367220 1,1,0,1,1,2,3,3,4,5,7,7,10,11,15,17,22,25,32,37,46,53,65,75,90,105,
%T A367220 124,143,168,193,224,258,297,340,390,446,509,580,660,751,852,967,1095,
%U A367220 1240,1401,1584,1786,2015,2269,2554,2869,3226,3617,4056,4541,5084
%N A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.
%C A367220 The non-strict version is A367218.
%e A367220 The a(3) = 1 through a(10) = 7 strict partitions:
%e A367220   (2,1)  (3,1)  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
%e A367220                 (4,1)  (5,1)    (6,1)    (7,1)    (8,1)    (9,1)
%e A367220                        (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
%e A367220                                          (5,2,1)  (5,3,1)  (5,4,1)
%e A367220                                                   (6,2,1)  (6,3,1)
%e A367220                                                            (7,2,1)
%e A367220                                                            (4,3,2,1)
%t A367220 combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t A367220 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]
%Y A367220 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 A367220                sum-full   sum-free   comb-full  comb-free
%Y A367220               -------------------------------------------
%Y A367220   partitions:  A367212    A367213    A367218    A367219
%Y A367220   strict:      A367214    A367215    A367220*   A367221
%Y A367220   subsets:     A367216    A367217    A367222    A367223
%Y A367220   ranks:       A367224    A367225    A367226    A367227
%Y A367220 A000041 counts integer partitions, strict A000009.
%Y A367220 A002865 counts partitions whose length is a part, complement A229816.
%Y A367220 A188431 counts complete strict partitions, incomplete A365831.
%Y A367220 A240855 counts strict partitions whose length is a part, complement A240861.
%Y A367220 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A367220 A365046 counts combination-full subsets, differences of A364914.
%Y A367220 Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.
%K A367220 nonn
%O A367220 0,6
%A A367220 _Gus Wiseman_, Nov 14 2023