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A375127 The anti-run-leader transformation for standard compositions.

%I A375127 #7 Aug 03 2024 07:12:16
%S A375127 0,1,2,3,4,2,1,7,8,4,10,5,1,1,3,15,16,8,4,9,2,10,2,11,1,1,6,3,3,3,7,
%T A375127 31,32,16,8,17,36,4,4,19,2,2,42,21,2,2,5,23,1,1,1,3,1,6,1,7,3,3,14,7,
%U A375127 7,7,15,63,64,32,16,33,8,8,8,35,4,36,18,9,4,4,9
%N A375127 The anti-run-leader transformation for standard compositions.
%C A375127 The a(n)-th composition in standard order lists the leaders of anti-runs of the n-th composition in standard order.
%C A375127 The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%C A375127 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 A375127 Does this sequence contain all nonnegative integers?
%H A375127 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>.
%H A375127 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375127 A000120(a(n)) = A333381(n).
%F A375127 A065120(a(n)) = A065120(n).
%F A375127 A070939(a(n)) = A374516(n).
%e A375127 The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2). This is the 42nd composition in standard order, so a(346) = 42.
%t A375127 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375127 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A375127 Table[stcinv[First/@Split[stc[n],UnsameQ]],{n,0,100}]
%Y A375127 Positions of elements of A233564 are A374638, counted by A374518.
%Y A375127 Positions of elements of A272919 are A374519, counted by A374517.
%Y A375127 Ranks of rows of A374515.
%Y A375127 A011782 counts compositions.
%Y A375127 A238130, A238279, A333755 count compositions by number of runs.
%Y A375127 All of the following pertain to compositions in standard order:
%Y A375127 - Length is A000120.
%Y A375127 - Sum is A029837(n+1) = A070939(n).
%Y A375127 - Parts are listed by A066099.
%Y A375127 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375127 - Run-length transform is A333627.
%Y A375127 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A375127 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A375127 - Run-sum transform is A353847.
%Y A375127 Six types of runs:
%Y A375127 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375127 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375127 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A375127 Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374516, A374521, A374685, A374698, A374701, A374767.
%K A375127 nonn
%O A375127 0,3
%A A375127 _Gus Wiseman_, Aug 02 2024