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%I #9 Sep 09 2022 15:25:03
%S 0,1,2,3,4,5,6,7,8,9,10,12,15,16,17,18,19,20,21,23,24,26,28,30,31,32,
%T 33,34,35,36,37,38,40,41,42,43,44,47,48,50,51,52,55,56,57,58,61,62,63,
%U 64,65,66,67,68,69,70,71,72,73,74,79,80,81,84,85,86,87,88
%N Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.
%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%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).
%e The terms together with their binary expansions and corresponding compositions begin:
%e 0: 0 ()
%e 1: 1 (1)
%e 2: 10 (2)
%e 3: 11 (1,1)
%e 4: 100 (3)
%e 5: 101 (2,1)
%e 6: 110 (1,2)
%e 7: 111 (1,1,1)
%e 8: 1000 (4)
%e 9: 1001 (3,1)
%e 10: 1010 (2,2)
%e 12: 1100 (1,3)
%e 15: 1111 (1,1,1,1)
%e 16: 10000 (5)
%e 17: 10001 (4,1)
%e 18: 10010 (3,2)
%e 19: 10011 (3,1,1)
%e 20: 10100 (2,3)
%e 21: 10101 (2,2,1)
%e 23: 10111 (2,1,1,1)
%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,100],UnsameQ@@Total/@Split[stc[#]]&]
%Y The version for runs in binary expansion is A175413.
%Y The version for parts instead of run-sums is A233564, counted A032020.
%Y The version for run-lengths instead of run-sums is A351596, counted A329739.
%Y The version for runs instead of run-sums is A351290, counted by A351013.
%Y The version for partitions is A353838, counted A353837, complement A353839.
%Y The equal instead of distinct version is A353848, counted by A353851.
%Y These compositions are counted by A353850.
%Y The weak version (rucksack compositions) is A354581, counted by A354580.
%Y A003242 counts anti-run compositions, ranked by A333489.
%Y A005811 counts runs in binary expansion.
%Y A011782 counts compositions.
%Y A242882 counts composition with distinct multiplicities, partitions A098859.
%Y A304442 counts partitions with all equal run-sums.
%Y A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353853-A353859 pertain to composition run-sum trajectory.
%Y A353864 counts rucksack partitions, perfect A353865.
%Y A353929 counts distinct runs in binary expansion, firsts A353930.
%Y Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 31 2022