A367221 - OEIS

%I #7 Nov 15 2023 08:23:04

%S 0,0,1,1,1,1,1,2,2,3,3,5,5,7,7,10,10,13,14,17,18,23,24,29,32,37,41,49,

%T 54,63,72,82,93,108,122,139,159,180,204,231,261,293,331,370,415,464,

%U 518,575,641,710,789,871,965,1064,1177,1294,1428,1569,1729,1897

%N Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

%C The non-strict version is A367219.

%e The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):

%e   2  3  4  5  6  7   8   9   A   B    C    D    E    F    G

%e                  43  53  54  64  65   75   76   86   87   97

%e                          63  73  74   84   85   95   96   A6

%e                                  83   93   94   A4   A5   B5

%e                                  542  642  A3   B3   B4   C4

%e                                            652  752  C3   D3

%e                                            742  842  654  754

%e                                                      762  862

%e                                                      852  952

%e                                                      942  A42

%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[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]=={}&]], {n,0,30}]

%Y 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                sum-full   sum-free   comb-full  comb-free

%Y               -------------------------------------------

%Y   partitions:  A367212    A367213    A367218    A367219

%Y   strict:      A367214    A367215    A367220    A367221*

%Y   subsets:     A367216    A367217    A367222    A367223

%Y   ranks:       A367224    A367225    A367226    A367227

%Y A000041 counts integer partitions, strict A000009.

%Y A002865 counts partitions whose length is a part, complement A229816.

%Y A124506 appears to count combination-free subsets, differences of A326083.

%Y A188431 counts complete strict partitions, incomplete A365831.

%Y A240855 counts strict partitions whose length is a part, complement A240861.

%Y A364272 counts sum-full strict partitions, sum-free A364349.

%Y Triangles:

%Y A008284 counts partitions by length, strict A008289.

%Y A046663 counts partitions of n without a subset-sum k, strict A365663.

%Y A365541 counts subsets containing two distinct elements summing to k.

%Y A365658 counts partitions by number of subset-sums, strict A365832.

%Y Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.

%K nonn

%O 0,8

%A _Gus Wiseman_, Nov 14 2023