cp's OEIS Frontend

A351596 Numbers k such that the k-th composition in standard order has all distinct run-lengths.

%I A351596 #7 Mar 13 2022 19:00:58
%S A351596 0,1,2,3,4,7,8,10,11,14,15,16,19,21,23,26,28,30,31,32,35,36,39,42,47,
%T A351596 56,60,62,63,64,67,71,73,74,79,84,85,87,95,100,106,112,119,120,122,
%U A351596 123,124,126,127,128,131,135,136,138,143,146,159,164,168,170,171
%N A351596 Numbers k such that the k-th composition in standard order has all distinct run-lengths.
%C A351596 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.
%e A351596 The terms together with their binary expansions and corresponding compositions begin:
%e A351596    0:      0  ()
%e A351596    1:      1  (1)
%e A351596    2:     10  (2)
%e A351596    3:     11  (1,1)
%e A351596    4:    100  (3)
%e A351596    7:    111  (1,1,1)
%e A351596    8:   1000  (4)
%e A351596   10:   1010  (2,2)
%e A351596   11:   1011  (2,1,1)
%e A351596   14:   1110  (1,1,2)
%e A351596   15:   1111  (1,1,1,1)
%e A351596   16:  10000  (5)
%e A351596   19:  10011  (3,1,1)
%e A351596   21:  10101  (2,2,1)
%e A351596   23:  10111  (2,1,1,1)
%t A351596 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A351596 Select[Range[0,100],UnsameQ@@Length/@Split[stc[#]]&]
%Y A351596 The version using binary expansions is A044813.
%Y A351596 The version for Heinz numbers and prime multiplicities is A130091.
%Y A351596 These compositions are counted by A329739, normal A329740.
%Y A351596 The version for runs instead of run-lengths is A351290, counted by A351013.
%Y A351596 A005811 counts runs in binary expansion, distinct A297770.
%Y A351596 A011782 counts integer compositions.
%Y A351596 A085207 represents concatenation of standard compositions, reverse A085208.
%Y A351596 A333489 ranks anti-runs, complement A348612.
%Y A351596 A345167 ranks alternating compositions, counted by A025047.
%Y A351596 A351204 counts partitions where every permutation has all distinct runs.
%Y A351596 Counting words with all distinct run-lengths:
%Y A351596 - A032020 = binary expansions, for runs A351018.
%Y A351596 - A351017 = binary words, for runs A351016.
%Y A351596 - A351292 = patterns, for runs A351200.
%Y A351596 Selected statistics of standard compositions (A066099, A228351):
%Y A351596 - Length is A000120.
%Y A351596 - Sum is A070939.
%Y A351596 - Runs are counted by A124767, distinct A351014.
%Y A351596 - Heinz number is A333219.
%Y A351596 - Number of distinct parts is A334028.
%Y A351596 Cf. A098859, A106356, A175413, A238279, A242882, A328592, A329745, A333628, A350952, A351015, A351202.
%K A351596 nonn
%O A351596 1,3
%A A351596 _Gus Wiseman_, Feb 24 2022