# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a353864 Showing 1-1 of 1 %I A353864 #13 May 25 2022 09:12:49 %S A353864 1,1,2,3,4,6,8,11,14,19,25,33,39,51,65,82,101,126,154,191,232,284,343, %T A353864 416,496,600,716,855,1018,1209,1430,1691,1991,2345,2747,3224,3762, %U A353864 4393,5116,5946,6897,7998,9257,10696,12336,14213,16343,18781,21538,24687 %N A353864 Number of rucksack partitions of n: every consecutive constant subsequence has a different sum. %C A353864 In a knapsack partition (A108917), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short. Another variation of knapsack partitions is A325862. %e A353864 The a(0) = 1 through a(7) = 11 partitions: %e A353864 () (1) (2) (3) (4) (5) (6) (7) %e A353864 (11) (21) (22) (32) (33) (43) %e A353864 (111) (31) (41) (42) (52) %e A353864 (1111) (221) (51) (61) %e A353864 (311) (222) (322) %e A353864 (11111) (321) (331) %e A353864 (411) (421) %e A353864 (111111) (511) %e A353864 (2221) %e A353864 (4111) %e A353864 (1111111) %t A353864 msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]]; %t A353864 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Select[msubs[#],SameQ@@#&]&]],{n,0,30}] %Y A353864 Knapsack partitions are counted by A108917, ranked by A299702. %Y A353864 The strong case is A353838, counted by A353837, complement A353839. %Y A353864 The perfect case is A353865, ranked by A353867. %Y A353864 These partitions are ranked by A353866. %Y A353864 A000041 counts partitions, strict A000009. %Y A353864 A300273 ranks collapsible partitions, counted by A275870. %Y A353864 A304442 counts partitions with all equal run-sums, ranked by A353833. %Y A353864 A353832 represents the operation of taking run-sums of a partition. %Y A353864 A353836 counts partitions by number of distinct run-sums. %Y A353864 A353840-A353846 pertain to partition run-sum trajectory. %Y A353864 A353852 ranks compositions with all distinct run-sums, counted by A353850. %Y A353864 A353863 counts partitions whose weak run-sums cover an initial interval. %Y A353864 Cf. A008284, A018818, A225485, A325239, A325277, A325280, A325862, A353834. %K A353864 nonn %O A353864 0,3 %A A353864 _Gus Wiseman_, May 23 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE