A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 4, 0, 0, 8, 3, 0, 10, 4, 4, 15, 4, 8, 24, 7, 8, 42, 16, 10, 59, 31, 27, 87, 37, 52, 149, 62, 66, 233, 121, 111, 342, 207, 204, 531, 308, 351, 864, 487, 536, 1373, 864, 865, 2057, 1440, 1509, 3232

Offset: 0

Author

Gus Wiseman, May 16 2022

Keywords

• nonn

Examples

			The a(n) compositions for selected n:
  n=16:   n=18:     n=20:    n=21:      n=24:
----------------------------------------------------
  (4444)  (666)     (5555)   (777)      (888)
          (333333)  (44444)  (333444)   (6666)
                             (444333)   (333555)
                             (3333333)  (444444)
                                        (555333)
                                        (3333444)
                                        (4443333)
                                        (33333333)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Allowing any multiplicities gives A078012, partitions A008483.

The version for no (instead of all) parts or run-lengths > 2 is A137200.

Allowing any parts gives A353400, partitions A100405.

The version for partitions is A353501, ranked by A353502.

The version for > 1 instead of > 2 is A353508, partitions A339222.

A003242 counts anti-run compositions, ranked by A333489.

A008466 counts compositions with some part > 2.

A011782 counts compositions.

A114901 counts compositions with no runs of length 1, ranked by A353427.

A128695 counts compositions with no run-lengths > 2.

A261983 counts non-anti-run compositions.

A335464 counts compositions with a run-length > 2.

Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.

Programs

Maple

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 18 2022

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

Extensions

a(26)-a(66) from Alois P. Heinz, May 17 2022