A344614 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.

1, 1, 2, 4, 8, 16, 30, 58, 110, 209, 397, 753, 1429, 2711, 5143, 9757, 18511, 35117, 66621, 126389, 239781, 454897, 863010, 1637260, 3106138, 5892821, 11179603, 21209446, 40237641, 76337091, 144823431, 274752731, 521249018, 988891100, 1876081530, 3559220898, 6752400377

Offset

0,3

Comments

These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.

Links

Table of n, a(n) for n=0..36.

Example

The a(6) = 30 compositions are:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (24)  (132)  (1122)  (11121)
       (33)  (141)  (1131)  (11211)
       (42)  (213)  (1212)  (12111)
       (51)  (222)  (1221)  (21111)
             (231)  (1311)
             (312)  (2112)
             (411)  (2121)
                    (2211)
                    (3111)
Missing are: (123), (321).

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, y_, z_, ___}/; x<y<z||x>y>z]&]], {n, 0, 15}]

Crossrefs

A001250 counts alternating permutations.

A005649 counts anti-run patterns.

A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

A106356 counts compositions by number of maximal anti-runs.

A114901 counts compositions where each part is adjacent to an equal part.

A325534 counts separable partitions.

A325535 counts inseparable partitions.

A344604 counts wiggly compositions with twins.

A344605 counts wiggly patterns with twins.

A344606 counts wiggly permutations of prime factors with twins.

Counting compositions by patterns:

- A003242 avoiding (1,1) adjacent.

- A011782 no conditions.

- A106351 avoiding (1,1) adjacent by sum and length.

- A128695 avoiding (1,1,1) adjacent.

- A128761 avoiding (1,2,3).

- A232432 avoiding (1,1,1).

- A335456 all patterns.

- A335457 all patterns adjacent.

- A335514 matching (1,2,3).

- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.

- A344614 avoiding (1,2,3) and (3,2,1) adjacent.

- A344615 weakly avoiding (1,2,3) adjacent.

Cf. A000041, A006330, A008965, A049774, A056986, A238279/A333755, A333213, A335515, A344612, A344652.

Sequence in context: A164180 A164179 A027559 * A337664 A135492 A164191

Adjacent sequences: A344611 A344612 A344613 * A344615 A344616 A344617

Keyword

nonn

Author

Gus Wiseman, May 27 2021

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

Status

approved