cp's OEIS Frontend

A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

%I A351290 #8 Feb 13 2022 09:52:38
%S A351290 0,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,23,24,26,27,28,
%T A351290 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,47,48,50,51,52,55,56,
%U A351290 57,58,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,78
%N A351290 Numbers k such that the k-th composition in standard order has all distinct runs.
%C A351290 The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%H A351290 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351290 The terms together with their binary expansions and corresponding compositions begin:
%e A351290    0:      0  ()
%e A351290    1:      1  (1)
%e A351290    2:     10  (2)
%e A351290    3:     11  (1,1)
%e A351290    4:    100  (3)
%e A351290    5:    101  (2,1)
%e A351290    6:    110  (1,2)
%e A351290    7:    111  (1,1,1)
%e A351290    8:   1000  (4)
%e A351290    9:   1001  (3,1)
%e A351290   10:   1010  (2,2)
%e A351290   11:   1011  (2,1,1)
%e A351290   12:   1100  (1,3)
%e A351290   14:   1110  (1,1,2)
%e A351290   15:   1111  (1,1,1,1)
%t A351290 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351290 Select[Range[0,100],UnsameQ@@Split[stc[#]]&]
%Y A351290 The version for Heinz numbers and prime multiplicities is A130091.
%Y A351290 The version using binary expansions is A175413, complement A351205.
%Y A351290 The version for run-lengths instead of runs is A329739.
%Y A351290 These compositions are counted by A351013.
%Y A351290 The complement is A351291.
%Y A351290 A005811 counts runs in binary expansion, distinct A297770.
%Y A351290 A011782 counts integer compositions.
%Y A351290 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351290 A085207 represents concatenation of standard compositions, reverse A085208.
%Y A351290 A333489 ranks anti-runs, complement A348612.
%Y A351290 A345167 ranks alternating compositions, counted by A025047.
%Y A351290 A351204 counts partitions where every permutation has all distinct runs.
%Y A351290 Counting words with all distinct runs:
%Y A351290 - A351016 = binary words, for run-lengths A351017.
%Y A351290 - A351018 = binary expansions, for run-lengths A032020.
%Y A351290 - A351200 = patterns, for run-lengths A351292.
%Y A351290 - A351202 = permutations of prime factors.
%Y A351290 Selected statistics of standard compositions:
%Y A351290 - Length is A000120.
%Y A351290 - Parts are A066099, reverse A228351.
%Y A351290 - Sum is A070939.
%Y A351290 - Runs are counted by A124767, distinct A351014.
%Y A351290 - Heinz number is A333219.
%Y A351290 - Number of distinct parts is A334028.
%Y A351290 Selected classes of standard compositions:
%Y A351290 - Partitions are A114994, strict A333256.
%Y A351290 - Multisets are A225620, strict A333255.
%Y A351290 - Strict compositions are A233564.
%Y A351290 - Constant compositions are A272919.
%Y A351290 Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.
%K A351290 nonn
%O A351290 1,3
%A A351290 _Gus Wiseman_, Feb 10 2022