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

Author

Gus Wiseman, May 27 2021

Keywords

• nonn

Comments

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

Examples

			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).
		

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.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;xy>z]&]],{n,0,15}]

Extensions

More terms from Bert Dobbelaere, Jun 12 2021