%I #13 May 25 2022 09:12:49

%S 1,1,2,3,4,6,8,11,14,19,25,33,39,51,65,82,101,126,154,191,232,284,343,

%T 416,496,600,716,855,1018,1209,1430,1691,1991,2345,2747,3224,3762,

%U 4393,5116,5946,6897,7998,9257,10696,12336,14213,16343,18781,21538,24687

%N Number of rucksack partitions of n: every consecutive constant subsequence has a different sum.

%C 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 The a(0) = 1 through a(7) = 11 partitions:

%e () (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (111) (31) (41) (42) (52)

%e (1111) (221) (51) (61)

%e (311) (222) (322)

%e (11111) (321) (331)

%e (411) (421)

%e (111111) (511)

%e (2221)

%e (4111)

%e (1111111)

%t msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Select[msubs[#],SameQ@@#&]&]],{n,0,30}]

%Y Knapsack partitions are counted by A108917, ranked by A299702.

%Y The strong case is A353838, counted by A353837, complement A353839.

%Y The perfect case is A353865, ranked by A353867.

%Y These partitions are ranked by A353866.

%Y A000041 counts partitions, strict A000009.

%Y A300273 ranks collapsible partitions, counted by A275870.

%Y A304442 counts partitions with all equal run-sums, ranked by A353833.

%Y A353832 represents the operation of taking run-sums of a partition.

%Y A353836 counts partitions by number of distinct run-sums.

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y A353852 ranks compositions with all distinct run-sums, counted by A353850.

%Y A353863 counts partitions whose weak run-sums cover an initial interval.

%Y Cf. A008284, A018818, A225485, A325239, A325277, A325280, A325862, A353834.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 23 2022