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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

%I A351291 #6 Feb 13 2022 09:53:00
%S A351291 13,22,25,45,46,49,53,54,59,76,77,82,89,91,93,94,97,101,102,105,108,
%T A351291 109,110,115,118,141,148,150,153,156,162,165,166,173,177,178,180,181,
%U A351291 182,183,187,189,190,193,197,198,201,204,205,209,210,213,214,216,217
%N A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.
%C A351291 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 A351291 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351291 The terms together with their binary expansions and corresponding compositions begin:
%e A351291   13:     1101  (1,2,1)
%e A351291   22:    10110  (2,1,2)
%e A351291   25:    11001  (1,3,1)
%e A351291   45:   101101  (2,1,2,1)
%e A351291   46:   101110  (2,1,1,2)
%e A351291   49:   110001  (1,4,1)
%e A351291   53:   110101  (1,2,2,1)
%e A351291   54:   110110  (1,2,1,2)
%e A351291   59:   111011  (1,1,2,1,1)
%e A351291   76:  1001100  (3,1,3)
%e A351291   77:  1001101  (3,1,2,1)
%e A351291   82:  1010010  (2,3,2)
%e A351291   89:  1011001  (2,1,3,1)
%e A351291   91:  1011011  (2,1,2,1,1)
%e A351291   93:  1011101  (2,1,1,2,1)
%e A351291   94:  1011110  (2,1,1,1,2)
%t A351291 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351291 Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]
%Y A351291 The version for Heinz numbers of partitions is A130092, complement A130091.
%Y A351291 Normal multisets with a permutation of this type appear to be A283353.
%Y A351291 Partitions w/o permutations of this type are A351204, complement A351203.
%Y A351291 The version using binary expansions is A351205, complement A175413.
%Y A351291 The complement is A351290, counted by A351013.
%Y A351291 A005811 counts runs in binary expansion, distinct A297770.
%Y A351291 A011782 counts integer compositions.
%Y A351291 A044813 lists numbers whose binary expansion has all distinct run-lengths.
%Y A351291 A085207 represents concatenation of standard compositions, reverse A085208.
%Y A351291 A333489 ranks anti-runs, complement A348612, counted by A003242.
%Y A351291 A345167 ranks alternating compositions, counted by A025047.
%Y A351291 Counting words with all distinct runs:
%Y A351291 - A351016 = binary words, for run-lengths A351017.
%Y A351291 - A351018 = binary expansions, for run-lengths A032020.
%Y A351291 - A351200 = patterns, for run-lengths A351292.
%Y A351291 - A351202 = permutations of prime factors.
%Y A351291 Selected statistics of standard compositions (A066099, reverse A228351):
%Y A351291 - Length is A000120.
%Y A351291 - Sum is A070939.
%Y A351291 - Runs are counted by A124767, distinct A351014.
%Y A351291 - Heinz number is A333219.
%Y A351291 - Number of distinct parts is A334028.
%Y A351291 Selected classes of standard compositions:
%Y A351291 - Partitions are A114994, strict A333256.
%Y A351291 - Multisets are A225620, strict A333255.
%Y A351291 - Strict compositions are A233564.
%Y A351291 - Constant compositions are A272919.
%Y A351291 Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.
%K A351291 nonn
%O A351291 1,1
%A A351291 _Gus Wiseman_, Feb 12 2022