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A374253 Numbers k such that the k-th composition in standard order matches the patterns (1,2,1) or (2,1,2).

%I A374253 #7 Jul 14 2024 08:54:40
%S A374253 13,22,25,27,29,45,46,49,51,53,54,55,57,59,61,76,77,82,86,89,90,91,93,
%T A374253 94,97,99,101,102,103,105,107,108,109,110,111,113,115,117,118,119,121,
%U A374253 123,125,141,148,150,153,155,156,157,162,165,166,173,174,177,178
%N A374253 Numbers k such that the k-th composition in standard order matches the patterns (1,2,1) or (2,1,2).
%C A374253 Such a composition cannot be strict.
%C A374253 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.
%H A374253 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A374253 Equals A335466 \/ A335468.
%e A374253 The terms together with their standard compositions begin:
%e A374253   13: (1,2,1)
%e A374253   22: (2,1,2)
%e A374253   25: (1,3,1)
%e A374253   27: (1,2,1,1)
%e A374253   29: (1,1,2,1)
%e A374253   45: (2,1,2,1)
%e A374253   46: (2,1,1,2)
%e A374253   49: (1,4,1)
%e A374253   51: (1,3,1,1)
%e A374253   53: (1,2,2,1)
%e A374253   54: (1,2,1,2)
%e A374253   55: (1,2,1,1,1)
%e A374253   57: (1,1,3,1)
%e A374253   59: (1,1,2,1,1)
%e A374253   61: (1,1,1,2,1)
%e A374253   76: (3,1,3)
%e A374253   77: (3,1,2,1)
%e A374253   82: (2,3,2)
%e A374253   86: (2,2,1,2)
%e A374253   89: (2,1,3,1)
%t A374253 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374253 Select[Range[0,100],!UnsameQ@@First/@Split[stc[#]]&]
%Y A374253 Permutations of prime indices of this type are counted by A335460.
%Y A374253 Compositions of this type are counted by A335548.
%Y A374253 The complement is A374249, counted by A274174.
%Y A374253 The anti-run case is A374254.
%Y A374253 A003242 counts anti-run compositions, ranks A333489.
%Y A374253 A011782 counts compositions.
%Y A374253 A025047 counts wiggly compositions, ranks A345167.
%Y A374253 A066099 lists compositions in standard order.
%Y A374253 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A374253 A233564 ranks strict compositions, counted by A032020.
%Y A374253 A333755 counts compositions by number of runs.
%Y A374253 A335454 counts patterns matched by standard compositions.
%Y A374253 A335456 counts patterns matched by compositions.
%Y A374253 A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
%Y A374253 A335465 counts minimal patterns avoided by a standard composition.
%Y A374253 - A335470 counts (1,2,1)-matching compositions, ranks A335466.
%Y A374253 - A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
%Y A374253 - A335472 counts (2,1,2)-matching compositions, ranks A335468.
%Y A374253 - A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
%Y A374253 A373948 encodes run-compression using compositions in standard order.
%Y A374253 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374253 A373953 gives run-compressed sum of standard compositions, excess A373954.
%Y A374253 Cf. A106356, A124762, A238130, A238279, A261982, A333175, A333382, A333627, A335463, A335524, A335525.
%K A374253 nonn
%O A374253 1,1
%A A374253 _Gus Wiseman_, Jul 13 2024