cp's OEIS Frontend

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

%I A367221 #7 Nov 15 2023 08:23:04
%S A367221 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 A367221 54,63,72,82,93,108,122,139,159,180,204,231,261,293,331,370,415,464,
%U A367221 518,575,641,710,789,871,965,1064,1177,1294,1428,1569,1729,1897
%N A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.
%C A367221 The non-strict version is A367219.
%e A367221 The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
%e A367221   2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
%e A367221                  43  53  54  64  65   75   76   86   87   97
%e A367221                          63  73  74   84   85   95   96   A6
%e A367221                                  83   93   94   A4   A5   B5
%e A367221                                  542  642  A3   B3   B4   C4
%e A367221                                            652  752  C3   D3
%e A367221                                            742  842  654  754
%e A367221                                                      762  862
%e A367221                                                      852  952
%e A367221                                                      942  A42
%t A367221 combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t A367221 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]=={}&]], {n,0,30}]
%Y A367221 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 A367221                sum-full   sum-free   comb-full  comb-free
%Y A367221               -------------------------------------------
%Y A367221   partitions:  A367212    A367213    A367218    A367219
%Y A367221   strict:      A367214    A367215    A367220    A367221*
%Y A367221   subsets:     A367216    A367217    A367222    A367223
%Y A367221   ranks:       A367224    A367225    A367226    A367227
%Y A367221 A000041 counts integer partitions, strict A000009.
%Y A367221 A002865 counts partitions whose length is a part, complement A229816.
%Y A367221 A124506 appears to count combination-free subsets, differences of A326083.
%Y A367221 A188431 counts complete strict partitions, incomplete A365831.
%Y A367221 A240855 counts strict partitions whose length is a part, complement A240861.
%Y A367221 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A367221 Triangles:
%Y A367221 A008284 counts partitions by length, strict A008289.
%Y A367221 A046663 counts partitions of n without a subset-sum k, strict A365663.
%Y A367221 A365541 counts subsets containing two distinct elements summing to k.
%Y A367221 A365658 counts partitions by number of subset-sums, strict A365832.
%Y A367221 Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.
%K A367221 nonn
%O A367221 0,8
%A A367221 _Gus Wiseman_, Nov 14 2023