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A375123 Weakly increasing run-leader transformation for standard compositions.

%I A375123 #8 Aug 03 2024 07:12:39
%S A375123 0,1,2,1,4,5,1,1,8,9,2,5,1,3,1,1,16,17,18,9,2,5,5,5,1,3,1,3,1,3,1,1,
%T A375123 32,33,34,17,4,37,9,9,2,5,2,5,5,11,5,5,1,3,6,3,1,3,3,3,1,3,1,3,1,3,1,
%U A375123 1,64,65,66,33,68,69,17,17,4,9,18,37,9,19,9,9
%N A375123 Weakly increasing run-leader transformation for standard compositions.
%C A375123 The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.
%C A375123 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%C A375123 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 A375123 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 A375123 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375123 A000120(a(n)) = A124766(n).
%F A375123 A070939(a(n)) = A374630(n) for n > 0.
%F A375123 A065120(a(n)) = A065120(n).
%e A375123 The 813th composition in standard order is (1,3,2,1,2,1), with weakly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.
%t A375123 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375123 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A375123 Table[stcinv[First/@Split[stc[n],LessEqual]],{n,0,100}]
%Y A375123 Positions of elements of A233564 are A374768, counted by A374632.
%Y A375123 Positions of elements of A272919 are A374633, counted by A374631.
%Y A375123 Ranks of rows of A374629.
%Y A375123 The opposite version is A375124.
%Y A375123 The strict version is A375125.
%Y A375123 The strict opposite version is A375126.
%Y A375123 A011782 counts compositions.
%Y A375123 A238130, A238279, A333755 count compositions by number of runs.
%Y A375123 All of the following pertain to compositions in standard order:
%Y A375123 - Length is A000120.
%Y A375123 - Sum is A029837(n+1).
%Y A375123 - Leader is A065120.
%Y A375123 - Parts are listed by A066099.
%Y A375123 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375123 - Run-length transform is A333627, sum A070939.
%Y A375123 - Run-sum transformation is A353847.
%Y A375123 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A375123 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A375123 Six types of runs:
%Y A375123 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375123 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375123 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A375123 Cf. A189076, A238343, A333213, A373949, A374634, A374635, A374637, A374743, A374744, A375128.
%K A375123 nonn
%O A375123 0,3
%A A375123 _Gus Wiseman_, Aug 02 2024