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A375126 Strictly decreasing run-leader transformation for standard compositions.

%I A375126 #5 Aug 03 2024 07:12:20
%S A375126 0,1,2,3,4,2,6,7,8,4,10,5,12,6,14,15,16,8,4,9,20,10,10,11,24,12,26,13,
%T A375126 28,14,30,31,32,16,8,17,36,4,18,19,40,20,42,21,20,10,22,23,48,24,12,
%U A375126 25,52,26,26,27,56,28,58,29,60,30,62,63,64,32,16,33,8,8
%N A375126 Strictly decreasing run-leader transformation for standard compositions.
%C A375126 The a(n)-th composition in standard order lists the leaders of strictly decreasing runs in the n-th composition in standard order.
%C A375126 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%C A375126 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 A375126 Does this sequence contain all nonnegative integers?
%H A375126 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 A375126 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375126 A000120(a(n)) = A124769(n).
%F A375126 A065120(a(n)) = A065120(n).
%F A375126 A070939(a(n)) = A374758(n).
%e A375126 The 813th composition in standard order is (1,3,2,1,2,1), with strictly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.
%t A375126 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375126 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A375126 Table[stcinv[First/@Split[stc[n],Greater]],{n,0,100}]
%Y A375126 Positions of elements of A233564 are A374767, counted by A374761.
%Y A375126 Positions of elements of A272919 are A374759, counted by A374760.
%Y A375126 Ranks of rows of A374757 (row-sums A374758).
%Y A375126 The weak opposite version is A375123.
%Y A375126 The weak version is A375124.
%Y A375126 The opposite version is A375125.
%Y A375126 A011782 counts compositions.
%Y A375126 A238130, A238279, A333755 count compositions by number of runs.
%Y A375126 All of the following pertain to compositions in standard order:
%Y A375126 - Length is A000120.
%Y A375126 - Sum is A029837(n+1) = A070939(n).
%Y A375126 - Parts are listed by A066099.
%Y A375126 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375126 - Run-length transform is A333627.
%Y A375126 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A375126 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A375126 - Run-sum transformation is A353847.
%Y A375126 Six types of runs:
%Y A375126 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375126 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375126 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A375126 Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374634, A374685, A374744, A374766, A375128.
%K A375126 nonn
%O A375126 0,3
%A A375126 _Gus Wiseman_, Aug 02 2024