A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.

7, 11, 13, 14, 19, 21, 25, 26, 28, 35, 37, 41, 42, 49, 50, 52, 56, 67, 69, 73, 74, 81, 82, 84, 97, 98, 100, 104, 112, 131, 133, 137, 138, 145, 146, 161, 162, 164, 168, 193, 194, 196, 200, 208, 224, 259, 261, 265, 266, 273, 274, 289, 290, 292, 321, 322, 324

Offset: 1

Author

Gus Wiseman, Sep 07 2020

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

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:
      7: (1,1,1)     52: (1,2,3)    133: (5,2,1)
     11: (2,1,1)     56: (1,1,4)    137: (4,3,1)
     13: (1,2,1)     67: (5,1,1)    138: (4,2,2)
     14: (1,1,2)     69: (4,2,1)    145: (3,4,1)
     19: (3,1,1)     73: (3,3,1)    146: (3,3,2)
     21: (2,2,1)     74: (3,2,2)    161: (2,5,1)
     25: (1,3,1)     81: (2,4,1)    162: (2,4,2)
     26: (1,2,2)     82: (2,3,2)    164: (2,3,3)
     28: (1,1,3)     84: (2,2,3)    168: (2,2,4)
     35: (4,1,1)     97: (1,5,1)    193: (1,6,1)
     37: (3,2,1)     98: (1,4,2)    194: (1,5,2)
     41: (2,3,1)    100: (1,3,3)    196: (1,4,3)
     42: (2,2,2)    104: (1,2,4)    200: (1,3,4)
     49: (1,4,1)    112: (1,1,5)    208: (1,2,5)
     50: (1,3,2)    131: (6,1,1)    224: (1,1,6)
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence

Crossrefs

A337460 is the non-unimodal version.

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

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.

A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.

A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.

A001523 counts unimodal compositions.

A007052 counts unimodal patterns.

A011782 counts unimodal permutations.

A115981 counts non-unimodal compositions.

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

- Length is A000120.

- Triples are A014311, with strict case A337453.

- Sum is A070939.

- Runs are counted by A124767.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Heinz number is A333219.

- Combinatory separations are counted by A334030.

- Non-unimodal compositions are A335373.

- Non-co-unimodal compositions are A335374.

Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, 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

Complement of A335373 in A014311.