A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

1, 1, 2, 3, 6, 10, 17, 29, 50, 84, 143, 241, 408, 688, 1162, 1959, 3305, 5571, 9393, 15832, 26688, 44980, 75812, 127769, 215338, 362911, 611620, 1030758, 1737131, 2927556, 4933760, 8314754, 14012668, 23615198, 39798098, 67070686, 113032453, 190490542, 321028554

Offset

0,3

Comments

These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.

Links

Table of n, a(n) for n=0..38.

Example

The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)    (4)      (5)        (6)
       (1,1)  (1,2)  (1,3)    (1,4)      (1,5)
              (2,1)  (2,2)    (2,3)      (2,4)
                     (3,1)    (3,2)      (3,3)
                     (1,2,1)  (4,1)      (4,2)
                     (2,1,1)  (1,3,1)    (5,1)
                              (2,1,2)    (1,3,2)
                              (2,2,1)    (1,4,1)
                              (3,1,1)    (2,1,3)
                              (1,2,1,1)  (2,3,1)
                                         (3,1,2)
                                         (3,2,1)
                                         (4,1,1)
                                         (1,2,1,2)
                                         (1,3,1,1)
                                         (2,1,2,1)
                                         (2,2,1,1)

Mathematica

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

Crossrefs

The case of permutations is A049774.

The strict non-adjacent version is A102726.

The case of permutations of prime indices is A344652.

A001250 counts alternating permutations.

A005649 counts anti-run patterns.

A106356 counts compositions by number of maximal anti-runs.

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

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, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.

Sequence in context: A001610 A324015 A238777 * A245437 A285665 A135431

Adjacent sequences: A344612 A344613 A344614 * A344616 A344617 A344618

Keyword

nonn

Author

Gus Wiseman, May 27 2021

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

Status

approved