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A353864 Number of rucksack partitions of n: every consecutive constant subsequence has a different sum.

%I A353864 #17 Apr 04 2025 00:43:33
%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 4393,5116,5946,6897,7998,9257,10696,12336,14213,16343,18781,21538,24687,28253,32291,36876,42057
%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
%E A353864 a(50)-a(53) from _Robert Price_, Apr 03 2025