%I #8 Jun 06 2022 08:05:03

%S 1,1,1,2,1,2,2,2,2,3,2,2,2,3,2,5,2,3,4,3,2,4,3,3,4,4,3,4,3,4,5,5,4,6,

%T 4,6,5,4,5,6,5,6,7,6,5,9,6,6,7,6,8

%N Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

%C Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). A weak run-sum is the sum of any consecutive constant subsequence.

%C Do all positive integers appear only finitely many times in this sequence?

%e The a(n) compositions for n = 1, 3, 9, 15, 18:

%e (1) (21) (4311) (54321) (543321)

%e (111) (51111) (532221) (654111)

%e (111111111) (651111) (7611111)

%e (81111111) (111111111111111111)

%e (111111111111111)

%e For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).

%t norqQ[m_]:=Sort[m]==Range[0,Max[m]];

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

%t Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]

%Y Perfect partitions are counted by A002033, ranked by A325780.

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

%Y This is the complete case of A353864, ranked by A353866.

%Y These partitions are ranked by A353867.

%Y A000041 counts partitions, strict A000009.

%Y A275870 counts collapsible partitions, ranked by A300273.

%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 A353837 counts partitions with distinct run-sums, ranked by A353838.

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

%Y A353850 counts compositions with all distinct run-sums, ranked by A353852.

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

%Y Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jun 04 2022