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Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).
11

%I #11 Sep 01 2022 19:48:36

%S 0,1,2,3,4,5,6,7,8,10,11,13,14,15,16,18,20,21,22,23,26,27,29,30,31,32,

%T 36,37,38,41,42,43,44,45,46,47,50,52,53,54,55,58,59,61,62,63,64,68,72,

%U 74,75,77,78,82,83,84,85,86,87,89,90,91,92,93,94,95,101

%N Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

%C The k-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 k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The terms and their corresponding standard compositions begin:

%e 0: ()

%e 1: (1)

%e 2: (2)

%e 3: (1,1)

%e 4: (3)

%e 5: (2,1)

%e 6: (1,2)

%e 7: (1,1,1)

%e 8: (4)

%e 10: (2,2)

%e 11: (2,1,1)

%e 13: (1,2,1)

%e 14: (1,1,2)

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

%e 16: (5)

%e 18: (3,2)

%e 20: (2,3)

%e 21: (2,2,1)

%t nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];

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

%t Select[Range[0,100],nogapQ[stc[#]]&]

%Y See link for sequences related to standard compositions.

%Y An unordered version is A073491, complement A073492.

%Y These compositions are counted by A107428.

%Y The complement is A356842.

%Y The non-initial case is A356843, unordered A356845.

%Y A356230 ranks gapless factorization lengths, firsts A356603.

%Y A356233 counts factorizations into gapless numbers.

%Y A356844 ranks compositions with at least one 1.

%Y Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224/A356225.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 31 2022