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A351015 Smallest k such that the k-th composition in standard order has n distinct runs.

%I A351015 #12 Mar 17 2022 04:52:57
%S A351015 0,1,5,27,155,1655,18039,281975
%N A351015 Smallest k such that the k-th composition in standard order has n distinct runs.
%C A351015 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.
%C A351015 It would be very interesting to have a formula or general construction for a(n). - _Gus Wiseman_, Feb 12 2022
%H A351015 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351015 The terms together with their binary expansions and corresponding compositions begin:
%e A351015        0:                    0  ()
%e A351015        1:                    1  (1)
%e A351015        5:                  101  (2,1)
%e A351015       27:                11011  (1,2,1,1)
%e A351015      155:             10011011  (3,1,2,1,1)
%e A351015     1655:          11001110111  (1,3,1,1,2,1,1,1)
%e A351015    18039:      100011001110111  (4,1,3,1,1,2,1,1,1)
%e A351015   281975:  1000100110101110111  (4,3,1,2,2,1,1,2,1,1,1)
%t A351015 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351015 s=Table[Length[Union[Split[stc[n]]]],{n,0,1000}];
%t A351015 Table[Position[s,k][[1,1]]-1,{k,Union[s]}]
%Y A351015 The version for Heinz numbers and prime multiplicities is A006939.
%Y A351015 Counting not necessarily distinct runs gives A113835 (up to zero).
%Y A351015 Using binary expansions instead of standard compositions gives A350952.
%Y A351015 These are the positions of first appearances in A351014.
%Y A351015 A005811 counts runs in binary expansion, distinct A297770.
%Y A351015 A011782 counts integer compositions.
%Y A351015 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351015 A085207 represents concatenation of standard compositions, reverse A085208.
%Y A351015 A333489 ranks anti-runs, complement A348612.
%Y A351015 Counting words with all distinct runs:
%Y A351015 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351015 - A351016 = binary words, for run-lengths A351017.
%Y A351015 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351015 Selected statistics of standard compositions (A066099, reverse A228351):
%Y A351015 - Length is A000120.
%Y A351015 - Sum is A070939.
%Y A351015 - Runs are counted by A124767.
%Y A351015 - Number of distinct parts is A334028.
%Y A351015 Cf. A106356, A238279, A242882, A318928, A325545, A328592, A329745, A351201, A351204.
%K A351015 nonn,more
%O A351015 0,3
%A A351015 _Gus Wiseman_, Feb 10 2022