A356843 - OEIS

%I #8 Sep 01 2022 19:48:26

%S 2,4,8,10,16,18,20,32,36,42,64,68,72,74,82,84,128,136,146,148,164,170,

%T 256,264,272,274,276,290,292,296,298,324,328,330,338,340,512,528,548,

%U 580,584,586,594,596,658,660,676,682,1024,1040,1056,1092,1096,1098

%N Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

%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>

%F Complement of A333217 in A356841.

%e The terms together with their corresponding standard compositions begin:

%e     2: (2)

%e     4: (3)

%e     8: (4)

%e    10: (2,2)

%e    16: (5)

%e    18: (3,2)

%e    20: (2,3)

%e    32: (6)

%e    36: (3,3)

%e    42: (2,2,2)

%e    64: (7)

%e    68: (4,3)

%e    72: (3,4)

%e    74: (3,2,2)

%e    82: (2,3,2)

%e    84: (2,2,3)

%t nogapQ[m_]:=Or[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[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]

%Y See link for sequences related to standard compositions.

%Y A subset of A022340.

%Y These compositions are counted by A251729.

%Y The unordered version (using Heinz numbers of partitions) is A356845.

%Y A333217 ranks complete compositions.

%Y A356230 ranks gapless factorization lengths, firsts A356603.

%Y A356233 counts factorizations into gapless numbers.

%Y A356841 ranks gapless compositions, counted by A107428.

%Y A356842 ranks non-gapless compositions, counted by A356846.

%Y A356844 ranks compositions with at least one 1.

%Y Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 01 2022