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A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

%I A353852 #9 Sep 09 2022 15:25:03
%S A353852 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 A353852 33,34,35,36,37,38,40,41,42,43,44,47,48,50,51,52,55,56,57,58,61,62,63,
%U A353852 64,65,66,67,68,69,70,71,72,73,74,79,80,81,84,85,86,87,88
%N A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.
%C A353852 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 A353852 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 A353852 The terms together with their binary expansions and corresponding compositions begin:
%e A353852    0:        0  ()
%e A353852    1:        1  (1)
%e A353852    2:       10  (2)
%e A353852    3:       11  (1,1)
%e A353852    4:      100  (3)
%e A353852    5:      101  (2,1)
%e A353852    6:      110  (1,2)
%e A353852    7:      111  (1,1,1)
%e A353852    8:     1000  (4)
%e A353852    9:     1001  (3,1)
%e A353852   10:     1010  (2,2)
%e A353852   12:     1100  (1,3)
%e A353852   15:     1111  (1,1,1,1)
%e A353852   16:    10000  (5)
%e A353852   17:    10001  (4,1)
%e A353852   18:    10010  (3,2)
%e A353852   19:    10011  (3,1,1)
%e A353852   20:    10100  (2,3)
%e A353852   21:    10101  (2,2,1)
%e A353852   23:    10111  (2,1,1,1)
%t A353852 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353852 Select[Range[0,100],UnsameQ@@Total/@Split[stc[#]]&]
%Y A353852 The version for runs in binary expansion is A175413.
%Y A353852 The version for parts instead of run-sums is A233564, counted A032020.
%Y A353852 The version for run-lengths instead of run-sums is A351596, counted A329739.
%Y A353852 The version for runs instead of run-sums is A351290, counted by A351013.
%Y A353852 The version for partitions is A353838, counted A353837, complement A353839.
%Y A353852 The equal instead of distinct version is A353848, counted by A353851.
%Y A353852 These compositions are counted by A353850.
%Y A353852 The weak version (rucksack compositions) is A354581, counted by A354580.
%Y A353852 A003242 counts anti-run compositions, ranked by A333489.
%Y A353852 A005811 counts runs in binary expansion.
%Y A353852 A011782 counts compositions.
%Y A353852 A242882 counts composition with distinct multiplicities, partitions A098859.
%Y A353852 A304442 counts partitions with all equal run-sums.
%Y A353852 A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353852 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353852 A353864 counts rucksack partitions, perfect A353865.
%Y A353852 A353929 counts distinct runs in binary expansion, firsts A353930.
%Y A353852 Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.
%K A353852 nonn
%O A353852 0,3
%A A353852 _Gus Wiseman_, May 31 2022