This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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