cp's OEIS Frontend

A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

%I A374249 #8 Jul 13 2024 23:23:35
%S A374249 0,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,23,24,26,28,30,
%T A374249 31,32,33,34,35,36,37,38,39,40,41,42,43,44,47,48,50,52,56,58,60,62,63,
%U A374249 64,65,66,67,68,69,70,71,72,73,74,75,78,79,80,81,83,84,85
%N A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.
%C A374249 These are compositions avoiding the patterns (1,2,1) and (2,1,2).
%C A374249 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 A374249 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A374249 Equals A335467 /\ A335469.
%e A374249 The terms together with their standard compositions begin:
%e A374249    0: ()
%e A374249    1: (1)
%e A374249    2: (2)
%e A374249    3: (1,1)
%e A374249    4: (3)
%e A374249    5: (2,1)
%e A374249    6: (1,2)
%e A374249    7: (1,1,1)
%e A374249    8: (4)
%e A374249    9: (3,1)
%e A374249   10: (2,2)
%e A374249   11: (2,1,1)
%e A374249   12: (1,3)
%e A374249   14: (1,1,2)
%e A374249   15: (1,1,1,1)
%e A374249   16: (5)
%e A374249 See A374253 for the complement: 13, 22, 25, 27, 29, ...
%t A374249 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374249 Select[Range[0,100],UnsameQ@@First/@Split[stc[#]]&]
%Y A374249 The strict (also anti-run) case is A233564, counted by A032020.
%Y A374249 Compositions of this type are counted by A274174.
%Y A374249 Permutations of prime indices of this type are counted by A333175.
%Y A374249 The complement is A374253 (anti-run A374254), counted by A335548.
%Y A374249 A003242 counts anti-run compositions, ranks A333489.
%Y A374249 A011782 counts compositions.
%Y A374249 A066099 lists compositions in standard order.
%Y A374249 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A374249 A333755 counts compositions by number of runs.
%Y A374249 A335454 counts patterns matched by standard compositions.
%Y A374249 A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
%Y A374249 - A335470 counts (1,2,1)-matching compositions, ranks A335466.
%Y A374249 - A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
%Y A374249 - A335472 counts (2,1,2)-matching compositions, ranks A335468.
%Y A374249 - A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
%Y A374249 Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.
%K A374249 nonn
%O A374249 1,3
%A A374249 _Gus Wiseman_, Jul 13 2024