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A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).

%I A374254 #4 Jul 18 2024 14:26:17
%S A374254 13,22,25,45,49,54,76,77,82,89,97,101,102,105,108,109,141,148,150,153,
%T A374254 162,165,166,177,178,180,182,193,197,198,204,205,209,210,216,217,269,
%U A374254 278,280,281,297,300,301,305,306,308,310,322,325,326,332,333,353,354
%N A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).
%C A374254 Such a composition cannot be strict.
%C A374254 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 A374254 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A374254 Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A374254 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 A374254 Equals A333489 /\ A374253.
%e A374254 The terms together with their standard compositions begin:
%e A374254    13: (1,2,1)
%e A374254    22: (2,1,2)
%e A374254    25: (1,3,1)
%e A374254    45: (2,1,2,1)
%e A374254    49: (1,4,1)
%e A374254    54: (1,2,1,2)
%e A374254    76: (3,1,3)
%e A374254    77: (3,1,2,1)
%e A374254    82: (2,3,2)
%e A374254    89: (2,1,3,1)
%e A374254    97: (1,5,1)
%e A374254   101: (1,3,2,1)
%e A374254   102: (1,3,1,2)
%e A374254   105: (1,2,3,1)
%e A374254   108: (1,2,1,3)
%e A374254   109: (1,2,1,2,1)
%e A374254   141: (4,1,2,1)
%e A374254   148: (3,2,3)
%e A374254   150: (3,2,1,2)
%e A374254   153: (3,1,3,1)
%t A374254 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374254 Select[Range[0,100],Length[Split[stc[#]]] == Length[stc[#]]&&!UnsameQ@@First/@Split[stc[#]]&]
%Y A374254 Compositions of this type are counted by A285981.
%Y A374254 Permutations of prime indices of this type are counted by A335460.
%Y A374254 This is the anti-run complement case of A374249, counted by A274174.
%Y A374254 This is the anti-run case of A374253, counted by A335548.
%Y A374254 A003242 counts anti-run compositions, ranks A333489.
%Y A374254 A011782 counts compositions.
%Y A374254 A025047 counts wiggly compositions, ranks A345167.
%Y A374254 A066099 lists compositions in standard order.
%Y A374254 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A374254 A233564 ranks strict compositions, counted by A032020.
%Y A374254 A333755 counts compositions by number of runs.
%Y A374254 A335454 counts patterns matched by standard compositions.
%Y A374254 A335456 counts patterns matched by compositions.
%Y A374254 A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
%Y A374254 A335465 counts minimal patterns avoided by a standard composition.
%Y A374254 - A335470 counts (1,2,1)-matching compositions, ranks A335466.
%Y A374254 - A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
%Y A374254 - A335472 counts (2,1,2)-matching compositions, ranks A335468.
%Y A374254 - A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
%Y A374254 A373948 encodes run-compression using compositions in standard order.
%Y A374254 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374254 A373953 gives run-compressed sum of standard compositions, excess A373954.
%Y A374254 Cf. A106356, A124762, A238130, A238279, A261982, A333175, A333382, A333627, A335463, A335524, A335525.
%K A374254 nonn
%O A374254 1,1
%A A374254 _Gus Wiseman_, Jul 14 2024