A333149 Number of strict compositions of n that are neither increasing nor decreasing.

0, 0, 0, 0, 0, 0, 4, 4, 8, 12, 38, 42, 72, 98, 150, 298, 372, 542, 760, 1070, 1428, 2600, 3120, 4550, 6050, 8478, 10976, 15220, 23872, 29950, 41276, 55062, 74096, 97148, 129786, 167256, 256070, 314454, 429338, 556364, 749266, 955746, 1275016, 1618054

Offset

0,7

Comments

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.

Links

Table of n, a(n) for n=0..43.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) = A032020(n) - 2*A000009(n) + 1.

Example

The a(6) = 4 through a(9) = 12 compositions:
  (1,3,2)  (1,4,2)  (1,4,3)  (1,5,3)
  (2,1,3)  (2,1,4)  (1,5,2)  (1,6,2)
  (2,3,1)  (2,4,1)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (2,5,1)  (2,4,3)
                    (3,1,4)  (2,6,1)
                    (3,4,1)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (3,4,2)
                             (3,5,1)
                             (4,2,3)
                             (5,1,3)
                             (6,1,2)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@#&&!Greater@@#&&!Less@@#&]], {n, 0, 10}]

Crossrefs

The non-strict case is A332834.

The complement is counted by A333147.

Strict partitions are A000009.

Strict compositions are A032020.

Non-unimodal strict compositions are A072707.

Strict partitions with increasing or decreasing run-lengths are A333190.

Strict compositions with increasing or decreasing run-lengths are A333191.

Unimodal compositions are A001523, with strict case A072706.

Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.

Sequence in context: A002368 A299474 A022087 * A095294 A190100 A244421

Adjacent sequences: A333146 A333147 A333148 * A333150 A333151 A333152

Keyword

nonn

Author

Gus Wiseman, May 16 2020

Status

approved