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A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

%I A354908 #8 Jun 24 2022 17:19:57
%S A354908 1,2,3,4,7,8,10,11,14,15,16,31,32,36,39,42,43,46,47,58,59,60,62,63,64,
%T A354908 127,128,136,138,139,142,143,168,170,171,174,175,184,186,187,190,191,
%U A354908 232,234,235,238,239,248,250,251,254,255,256,292,295,316,319,484
%N A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.
%C A354908 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 A354908 If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.
%e A354908 The terms together with their corresponding compositions begin:
%e A354908   1:(1)  2:(2)   4:(3)     8:(4)     16:(5)      32:(6)
%e A354908          3:(11)  7:(111)  10:(22)    31:(11111)  36:(33)
%e A354908                           11:(211)               39:(3111)
%e A354908                           14:(112)               42:(222)
%e A354908                           15:(1111)              43:(2211)
%e A354908                                                  46:(2112)
%e A354908                                                  47:(21111)
%e A354908                                                  58:(1122)
%e A354908                                                  59:(11211)
%e A354908                                                  60:(1113)
%e A354908                                                  62:(11112)
%e A354908                                                  63:(111111)
%t A354908 repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
%t A354908 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A354908 Select[Range[0,100],MemberQ[repcams[stc[#]],{_}]&]
%Y A354908 The standard compositions used here are A066099, run-sums A353847/A353932.
%Y A354908 The version for Heinz numbers of partitions is A300273, counted by A275870.
%Y A354908 These compositions are counted by A353860.
%Y A354908 A003242 counts anti-run compositions, ranked by A333489, complement A261983.
%Y A354908 A011782 counts compositions.
%Y A354908 A124767 counts runs in standard compositions.
%Y A354908 A238279 and A333755 count compositions by number of runs.
%Y A354908 A334968 counts distinct sums of subsequences of standard compositions.
%Y A354908 A351014 counts distinct runs of standard compositions, firsts A351015.
%Y A354908 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354908 A354582 counts distinct partial runs of standard compositions, sums A354907.
%Y A354908 Cf. A005811, A029837, A072639, A124771, A274174, A318928, A333381, A353848, A353849, A353850.
%K A354908 nonn
%O A354908 1,2
%A A354908 _Gus Wiseman_, Jun 23 2022