cp's OEIS Frontend

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

%I A375139 #6 Aug 13 2024 09:09:27
%S A375139 26,50,53,58,90,98,100,101,106,107,114,117,122,154,164,178,181,186,
%T A375139 194,196,197,201,202,203,210,212,213,214,215,218,226,228,229,234,235,
%U A375139 242,245,250,282,306,309,314,324,329,346,354,356,357,362,363,370,373,378
%N A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.
%C A375139 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%C A375139 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 A375139 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A375139 The terms together with corresponding compositions begin:
%e A375139    26: (1,2,2)
%e A375139    50: (1,3,2)
%e A375139    53: (1,2,2,1)
%e A375139    58: (1,1,2,2)
%e A375139    90: (2,1,2,2)
%e A375139    98: (1,4,2)
%e A375139   100: (1,3,3)
%e A375139   101: (1,3,2,1)
%e A375139   106: (1,2,2,2)
%e A375139   107: (1,2,2,1,1)
%e A375139   114: (1,1,3,2)
%e A375139   117: (1,1,2,2,1)
%e A375139   122: (1,1,1,2,2)
%e A375139   154: (3,1,2,2)
%e A375139   164: (2,3,3)
%e A375139   178: (2,1,3,2)
%e A375139   181: (2,1,2,2,1)
%e A375139   186: (2,1,1,2,2)
%t A375139 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375139 Select[Range[0,100],!GreaterEqual@@First/@Split[stc[#],Less]&]
%Y A375139 For leaders of identical runs we have A335485.
%Y A375139 Ranked by positions of non-weakly decreasing rows in A374683.
%Y A375139 For identical leaders we have A374685, counted by A374686.
%Y A375139 The complement is counted by A374697.
%Y A375139 For distinct leaders we have A374698, counted by A374687.
%Y A375139 Compositions of this type are counted by A375135.
%Y A375139 Weakly increasing leaders: A375137, counts A374636, complement A189076.
%Y A375139 Interchanging weak/strict: A375295, counted by A375140, complement A188920.
%Y A375139 A003242 counts anti-run compositions, ranks A333489.
%Y A375139 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A375139 All of the following pertain to compositions in standard order:
%Y A375139 - Length is A000120.
%Y A375139 - Sum is A029837(n+1).
%Y A375139 - Leader is A065120.
%Y A375139 - Parts are listed by A066099.
%Y A375139 - Strict compositions are A233564.
%Y A375139 - Run-length transform is A333627, sum A070939.
%Y A375139 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A375139 - Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375139 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375139 - Run-rank: A375123, A375124, A375125, A375126, A375127.
%Y A375139 Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.
%K A375139 nonn
%O A375139 1,1
%A A375139 _Gus Wiseman_, Aug 12 2024