A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 43, 46, 47, 58, 59, 60, 62, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 171, 174, 175, 184, 186, 187, 190, 191, 232, 234, 235, 238, 239, 248, 250, 251, 254, 255, 256, 292, 295, 316, 319, 484

Offset

1,2

Comments

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.

If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.

Links

Table of n, a(n) for n=1..58.

Example

The terms together with their corresponding compositions begin:
  1:(1)  2:(2)   4:(3)     8:(4)     16:(5)      32:(6)
         3:(11)  7:(111)  10:(22)    31:(11111)  36:(33)
                          11:(211)               39:(3111)
                          14:(112)               42:(222)
                          15:(1111)              43:(2211)
                                                 46:(2112)
                                                 47:(21111)
                                                 58:(1122)
                                                 59:(11211)
                                                 60:(1113)
                                                 62:(11112)
                                                 63:(111111)

Mathematica

repcams[q_List]:=repcams[q]=Union[{q}, If[UnsameQ@@q, {}, Union@@repcams/@Union[Insert[Drop[q, #], Plus@@Take[q, #], First[#]]&/@Select[Tuples[Range[Length[q]], 2], And[Less@@#, SameQ@@Take[q, #]]&]]]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], MemberQ[repcams[stc[#]], {_}]&]

Crossrefs

The standard compositions used here are A066099, run-sums A353847/A353932.

The version for Heinz numbers of partitions is A300273, counted by A275870.

These compositions are counted by A353860.

A003242 counts anti-run compositions, ranked by A333489, complement A261983.

A011782 counts compositions.

A124767 counts runs in standard compositions.

A238279 and A333755 count compositions by number of runs.

A334968 counts distinct sums of subsequences of standard compositions.

A351014 counts distinct runs of standard compositions, firsts A351015.

A353853-A353859 pertain to composition run-sum trajectory.

A354582 counts distinct partial runs of standard compositions, sums A354907.

Cf. A005811, A029837, A072639, A124771, A274174, A318928, A333381, A353848, A353849, A353850.

Sequence in context: A089190 A306677 A351596 * A353857 A353848 A330722

Adjacent sequences: A354905 A354906 A354907 * A354909 A354910 A354911

Keyword

nonn

Author

Gus Wiseman, Jun 23 2022

Status

approved