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A375124 Weakly decreasing run-leader transformation for standard compositions.

%I A375124 #5 Aug 03 2024 07:12:29
%S A375124 0,1,2,1,4,2,6,1,8,4,2,2,12,6,6,1,16,8,4,4,20,2,10,2,24,12,6,6,12,6,6,
%T A375124 1,32,16,8,8,4,4,18,4,40,20,2,2,20,10,10,2,48,24,12,12,52,6,26,6,24,
%U A375124 12,6,6,12,6,6,1,64,32,16,16,8,8,34,8,72,4,4,4,36
%N A375124 Weakly decreasing run-leader transformation for standard compositions.
%C A375124 The a(n)-th composition in standard order lists the leaders of weakly decreasing runs in the n-th composition in standard order.
%C A375124 The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
%C A375124 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 A375124 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 A375124 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375124 A000120(a(n)) = A124765(n).
%F A375124 A065120(a(n)) = A065120(n).
%F A375124 A070939(a(n)) = A374741(n).
%e A375124 The 813th composition in standard order is (1,3,2,1,2,1), with weakly 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 A375124 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375124 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A375124 Table[stcinv[First/@Split[stc[n],GreaterEqual]],{n,0,100}]
%Y A375124 Positions of elements of A233564 are A374701, counted by A374743.
%Y A375124 Positions of elements of A272919 are A374744, counted by A374742.
%Y A375124 Ranks of rows of A374740.
%Y A375124 The opposite version is A375123.
%Y A375124 The strict version is A375126.
%Y A375124 The strict opposite version is A375125.
%Y A375124 A011782 counts compositions.
%Y A375124 A238130, A238279, A333755 count compositions by number of runs.
%Y A375124 All of the following pertain to compositions in standard order:
%Y A375124 - Length is A000120.
%Y A375124 - Sum is A029837(n+1) = A070939(n).
%Y A375124 - Parts are listed by A066099.
%Y A375124 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375124 - Run-length transform is A333627, sum A070939.
%Y A375124 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A375124 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A375124 - Run-sum transformation is A353847.
%Y A375124 Six types of runs:
%Y A375124 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375124 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375124 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A375124 Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.
%K A375124 nonn
%O A375124 0,3
%A A375124 _Gus Wiseman_, Aug 02 2024