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A351014 Number of distinct runs in the n-th composition in standard order.

%I A351014 #14 Feb 10 2022 20:19:16
%S A351014 0,1,1,1,1,2,2,1,1,2,1,2,2,2,2,1,1,2,2,2,2,2,2,2,2,2,2,3,2,3,2,1,1,2,
%T A351014 2,2,1,3,3,2,2,3,1,2,3,2,2,2,2,2,3,3,3,2,2,3,2,3,2,2,2,3,2,1,1,2,2,2,
%U A351014 2,3,3,2,2,2,2,3,2,3,3,2,2,3,2,3,2,2,3
%N A351014 Number of distinct runs in the n-th composition in standard order.
%C A351014 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 A351014 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351014 The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.
%t A351014 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351014 Table[Length[Union[Split[stc[n]]]],{n,0,100}]
%Y A351014 Counting not necessarily distinct runs gives A124767.
%Y A351014 Using binary expansions instead of standard compositions gives A297770.
%Y A351014 Positions of first appearances are A351015.
%Y A351014 A005811 counts runs in binary expansion.
%Y A351014 A011782 counts integer compositions.
%Y A351014 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351014 A085207 represents concatenation of standard compositions, reverse A085208.
%Y A351014 A333489 ranks anti-runs, complement A348612.
%Y A351014 A345167 ranks alternating compositions, counted by A025047.
%Y A351014 A351204 counts partitions where every permutation has all distinct runs.
%Y A351014 Counting words with all distinct runs:
%Y A351014 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351014 - A351016 = binary words, for run-lengths A351017.
%Y A351014 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351014 - A351200 = patterns, for run-lengths A351292.
%Y A351014 - A351202 = permutations of prime factors.
%Y A351014 Selected statistics of standard compositions:
%Y A351014 - Length is A000120.
%Y A351014 - Sum is A070939.
%Y A351014 - Heinz number is A333219.
%Y A351014 - Number of distinct parts is A334028.
%Y A351014 Selected classes of standard compositions:
%Y A351014 - Partitions are A114994, strict A333256.
%Y A351014 - Multisets are A225620, strict A333255.
%Y A351014 - Strict compositions are A233564.
%Y A351014 - Constant compositions are A272919.
%Y A351014 Cf. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.
%K A351014 nonn
%O A351014 0,6
%A A351014 _Gus Wiseman_, Feb 07 2022