A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 79, 83, 84, 85, 86, 87, 90, 91, 94, 95, 99, 100, 103, 106, 111, 112, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Author

Gus Wiseman, Dec 15 2021

Keywords

• nonn

Comments

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

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.

This sequence ranks compositions that are weakly but not strongly alternating.

Examples

			The terms and corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)
		

Links

• Wikipedia, Alternating permutation

Crossrefs

Partitions of this type are counted by A349795, ranked by A350137.

Permutations of prime indices of this type are counted by A349798.

These compositions are counted by A349800.

A001250 = alternating permutations, ranked by A349051, complement A348615.

A003242 = Carlitz (anti-run) compositions, ranked by A333489.

A025047/A025048/A025049 = alternating compositions, ranked by A345167.

A261983 = non-anti-run compositions, ranked by A348612.

A345164 = alternating permutations of prime indices, with twins A344606.

A345165 = partitions without an alternating permutation, ranked by A345171.

A345170 = partitions with an alternating permutation, ranked by A345172.

A345166 = separable partitions with no alternations, ranked by A345173.

A345192 = non-alternating compositions, ranked by A345168.

A345195 = non-alternating anti-run compositions, ranked by A345169.

A349052/A129852/A129853 = weakly alternating compositions.

A349053 = non-weakly alternating compositions, ranked by A349057.

A349056 = weak alternations of prime indices, complement A349797.

A349060 = weak alternations of partitions, complement A349061.

Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],(whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#],{_,x_,x_,_}]&]

Formula

Equals A345168 \ A349057 = A348612 \ A349057.