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A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

%I A354580 #24 Sep 11 2023 15:53:18
%S A354580 1,1,2,4,6,12,22,39,68,125,227,402,710,1280,2281,4040,7196,12780,
%T A354580 22623,40136,71121,125863,222616,393305,695059,1227990,2167059,
%U A354580 3823029,6743268,11889431,20955548,36920415,65030404,114519168,201612634,354849227
%N A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.
%C A354580 We define a partial run of a sequence to be any contiguous constant subsequence. The term rucksack is short for run-knapsack.
%H A354580 Max Alekseyev, <a href="/A354580/b354580.txt">Table of n, a(n) for n = 0..65</a>
%e A354580 The a(0) = 1 through a(5) = 12 compositions:
%e A354580   ()  (1)  (2)    (3)      (4)        (5)
%e A354580            (1,1)  (1,2)    (1,3)      (1,4)
%e A354580                   (2,1)    (2,2)      (2,3)
%e A354580                   (1,1,1)  (3,1)      (3,2)
%e A354580                            (1,2,1)    (4,1)
%e A354580                            (1,1,1,1)  (1,1,3)
%e A354580                                       (1,2,2)
%e A354580                                       (1,3,1)
%e A354580                                       (2,1,2)
%e A354580                                       (2,2,1)
%e A354580                                       (3,1,1)
%e A354580                                       (1,1,1,1,1)
%t A354580 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],UnsameQ@@Total/@Union@@Subsets/@Split[#]&]],{n,0,15}]
%Y A354580 The knapsack version is A325676, ranked by A333223.
%Y A354580 The non-partial version for partitions is A353837, ranked by A353838 (complement A353839).
%Y A354580 The non-partial version is A353850, ranked by A353852.
%Y A354580 The version for partitions is A353864, ranked by A353866.
%Y A354580 The complete version for partitions is A353865, ranked by A353867.
%Y A354580 These compositions are ranked by A354581.
%Y A354580 A003242 counts anti-run compositions, ranked by A333489.
%Y A354580 A011782 counts compositions.
%Y A354580 A108917 counts knapsack partitions, ranked by A299702, strict A275972.
%Y A354580 A238279 and A333755 count compositions by number of runs.
%Y A354580 A275870 counts collapsible partitions, ranked by A300273.
%Y A354580 A353836 counts partitions by number of distinct run-sums.
%Y A354580 A353847 is the composition run-sum transformation.
%Y A354580 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A354580 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354580 A353860 counts collapsible compositions, ranked by A354908.
%Y A354580 Cf. A143823, A169942, A242882, A325545, A325680, A325682, A325685, A325687, A329739, A351017, A353849.
%K A354580 nonn
%O A354580 0,3
%A A354580 _Gus Wiseman_, Jun 13 2022
%E A354580 Terms a(16) onward from _Max Alekseyev_, Sep 10 2023