A337460 Numbers k such that the k-th composition in standard order is a non-unimodal triple.

22, 38, 44, 70, 76, 88, 134, 140, 148, 152, 176, 262, 268, 276, 280, 296, 304, 352, 518, 524, 532, 536, 552, 560, 592, 608, 704, 1030, 1036, 1044, 1048, 1064, 1072, 1096, 1104, 1120, 1184, 1216, 1408, 2054, 2060, 2068, 2072, 2088, 2096, 2120, 2128, 2144, 2192

Offset: 1

Author

Gus Wiseman, Sep 18 2020

Keywords

• nonn

Comments

These are triples matching the pattern (2,1,2), (3,1,2), or (2,1,3).

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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.

Examples

			The sequence together with the corresponding triples begins:
      22: (2,1,2)     296: (3,2,4)    1048: (6,1,4)
      38: (3,1,2)     304: (3,1,5)    1064: (5,2,4)
      44: (2,1,3)     352: (2,1,6)    1072: (5,1,5)
      70: (4,1,2)     518: (7,1,2)    1096: (4,3,4)
      76: (3,1,3)     524: (6,1,3)    1104: (4,2,5)
      88: (2,1,4)     532: (5,2,3)    1120: (4,1,6)
     134: (5,1,2)     536: (5,1,4)    1184: (3,2,6)
     140: (4,1,3)     552: (4,2,4)    1216: (3,1,7)
     148: (3,2,3)     560: (4,1,5)    1408: (2,1,8)
     152: (3,1,4)     592: (3,2,5)    2054: (9,1,2)
     176: (2,1,5)     608: (3,1,6)    2060: (8,1,3)
     262: (6,1,2)     704: (2,1,7)    2068: (7,2,3)
     268: (5,1,3)    1030: (8,1,2)    2072: (7,1,4)
     276: (4,2,3)    1036: (7,1,3)    2088: (6,2,4)
     280: (4,1,4)    1044: (6,2,3)    2096: (6,1,5)
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence
• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

A000212 counts unimodal triples.

A000217(n - 2) counts 3-part compositions.

A001399(n - 3) counts 3-part partitions.

A001399(n - 6) counts 3-part strict partitions.

A001399(n - 6)*2 counts non-unimodal 3-part strict compositions.

A001399(n - 6)*4 counts unimodal 3-part strict compositions.

A001399(n - 6)*6 counts 3-part strict compositions.

A001523 counts unimodal compositions.

A001840 counts non-unimodal triples.

A059204 counts non-unimodal permutations.

A115981 counts non-unimodal compositions.

A328509 counts non-unimodal patterns.

A337459 ranks unimodal triples.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Triples are A014311.

- Sum is A070939.

- Runs are counted by A124767.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Heinz number is A333219.

- Non-unimodal compositions are A335373.

- Non-co-unimodal compositions are A335374.

- Strict triples are A337453.

Cf. A007304, A014612, A069905, A072706, A156040, A211540, A227038, A332743, A337461, A337604.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&MatchQ[stc[#],{x_,y_,z_}/;x>y
				

Formula

Intersection of A014311 and A335373.