A351290 - OEIS

%I #8 Feb 13 2022 09:52:38

%S 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 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,47,48,50,51,52,55,56,

%U 57,58,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,78

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

%C 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 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>

%e The terms together with their binary expansions and corresponding compositions begin:

%e    0:      0  ()

%e    1:      1  (1)

%e    2:     10  (2)

%e    3:     11  (1,1)

%e    4:    100  (3)

%e    5:    101  (2,1)

%e    6:    110  (1,2)

%e    7:    111  (1,1,1)

%e    8:   1000  (4)

%e    9:   1001  (3,1)

%e   10:   1010  (2,2)

%e   11:   1011  (2,1,1)

%e   12:   1100  (1,3)

%e   14:   1110  (1,1,2)

%e   15:   1111  (1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],UnsameQ@@Split[stc[#]]&]

%Y The version for Heinz numbers and prime multiplicities is A130091.

%Y The version using binary expansions is A175413, complement A351205.

%Y The version for run-lengths instead of runs is A329739.

%Y These compositions are counted by A351013.

%Y The complement is A351291.

%Y A005811 counts runs in binary expansion, distinct A297770.

%Y A011782 counts integer compositions.

%Y A044813 lists numbers whose binary expansion has distinct run-lengths.

%Y A085207 represents concatenation of standard compositions, reverse A085208.

%Y A333489 ranks anti-runs, complement A348612.

%Y A345167 ranks alternating compositions, counted by A025047.

%Y A351204 counts partitions where every permutation has all distinct runs.

%Y Counting words with all distinct runs:

%Y - A351016 = binary words, for run-lengths A351017.

%Y - A351018 = binary expansions, for run-lengths A032020.

%Y - A351200 = patterns, for run-lengths A351292.

%Y - A351202 = permutations of prime factors.

%Y Selected statistics of standard compositions:

%Y - Length is A000120.

%Y - Parts are A066099, reverse A228351.

%Y - Sum is A070939.

%Y - Runs are counted by A124767, distinct A351014.

%Y - Heinz number is A333219.

%Y - Number of distinct parts is A334028.

%Y Selected classes of standard compositions:

%Y - Partitions are A114994, strict A333256.

%Y - Multisets are A225620, strict A333255.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

%K nonn

%O 1,3

%A _Gus Wiseman_, Feb 10 2022