%I #29 Aug 09 2023 11:47:51

%S 1,1,2,4,5,12,24,38,52,111,218,286,520,792,1358,2628,4155,5508,9246,

%T 13182,23480,45150,54540,94986,146016,213725,301104,478586,851506,

%U 1302234,1775482,2696942,3746894,6077784,8194466,12638334,21763463,28423976,45309850,62955524,94345474

%N Number of integer compositions of n with all distinct run-sums.

%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).

%H Joseph Likar, <a href="/A353850/b353850.txt">Table of n, a(n) for n = 0..120</a>

%e The a(0) = 1 through a(5) = 12 compositions:

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

%e (11) (12) (13) (14)

%e (21) (22) (23)

%e (111) (31) (32)

%e (1111) (41)

%e (113)

%e (122)

%e (221)

%e (311)

%e (1112)

%e (2111)

%e (11111)

%e For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]],{n,0,15}]

%Y For distinct parts instead of run-sums we have A032020.

%Y For distinct multiplicities instead of run-sums we have A242882.

%Y For distinct run-lengths instead of run-sums we have A329739, ptns A098859.

%Y For runs instead of run-sums we have A351013.

%Y For partitions we have A353837, ranked by A353838 (complement A353839).

%Y For equal instead of distinct run-sums we have A353851, ptns A304442.

%Y These compositions are ranked by A353852.

%Y The weak version (rucksack compositions) is A354580, ranked by A354581.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A005811 counts runs in binary expansion.

%Y A011782 counts compositions.

%Y A175413 lists numbers whose binary expansion has all distinct runs.

%Y A351014 counts distinct runs in standard compositions, firsts A351015.

%Y A353847 gives composition run-sum transformation.

%Y A353929 counts distinct runs in binary expansion, firsts A353930.

%Y Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 31 2022

%E Terms a(21) and onwards from _Joseph Likar_, Aug 04 2023