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A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

%I A354905 #14 Jun 22 2022 20:43:14
%S A354905 3,0,2,8,32,68,130,290,274,580,520,1298,2080,1096,2082,4168,2178,4164,
%T A354905 4386,35137,8328,8786,10274,8772,16712,20562,8712,16658,33320,41554,
%U A354905 33288,82210,34856,66628,33312,66642,34850,69704,140306,133448,69714,74308,133154
%N A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).
%C A354905 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 A354905 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 A354905 The terms and their corresponding compositions begin:
%e A354905      3: (1,1)
%e A354905      0: ()
%e A354905      2: (2)
%e A354905      8: (4)
%e A354905     32: (6)
%e A354905     68: (4,3)
%e A354905    130: (6,2)
%e A354905    290: (3,4,2)
%e A354905    274: (4,3,2)
%e A354905    580: (3,4,3)
%e A354905    520: (6,4)
%e A354905   1298: (2,4,3,2)
%e A354905 The inverse run-sum compositions for n = 2, 8, 32, 68, 130, 290:
%e A354905   (2)   (4)     (6)       (43)     (62)       (342)
%e A354905   (11)  (22)    (33)      (223)    (332)      (3411)
%e A354905         (1111)  (222)     (4111)   (611)      (11142)
%e A354905                 (111111)  (11113)  (3311)     (32211)
%e A354905                           (22111)  (22211)    (111411)
%e A354905                                    (1111112)  (311112)
%e A354905                                               (1112211)
%t A354905 nn=1000;
%t A354905 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A354905 antirunQ[y_]:=Length[Split[y]]==Length[y];
%t A354905 q=Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,nn}];
%t A354905 w=Last[Select[Table[Take[q+1,i],{i,nn}],Union[#]==Range[Max@@#]&]-1];
%t A354905 Table[Position[w,k][[1,1]]-1,{k,0,Max@@w}]
%Y A354905 The standard compositions used here are A066099, run-sums A353847/A353932.
%Y A354905 This is the position of the first appearance of n in A354578.
%Y A354905 A011782 counts compositions.
%Y A354905 A003242 counts anti-run compositions, ranked by A333489.
%Y A354905 A005811 counts runs in binary expansion.
%Y A354905 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A354905 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A354905 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A354905 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354905 A354904 lists positions of zeros in A354578, complement A354912.
%Y A354905 A354908 ranks collapsible compositions, counted by A353860.
%Y A354905 Cf. A000005, A029837, A124767, A238279/A333755, A274174, A333381, A353832, A353849, A353863, A354584.
%K A354905 nonn
%O A354905 0,1
%A A354905 _Gus Wiseman_, Jun 21 2022