A353400 Number of integer compositions of n with all run-lengths > 2.

1, 0, 0, 1, 1, 1, 2, 1, 2, 4, 4, 5, 11, 11, 14, 27, 29, 37, 61, 72, 97, 147, 181, 246, 368, 470, 632, 914, 1198, 1611, 2286, 3018, 4079, 5709, 7619, 10329, 14333, 19258, 26142, 36069, 48688, 66114, 90800, 122913, 167020, 228735, 310167, 421708, 576499, 782803

Offset: 0

Author

Gus Wiseman, May 15 2022

Keywords

• nonn

Examples

			The a(7) = 1 through a(12) = 11 compositions:
  1111111   2222       333         22222        1112222       444
            11111111   111222      1111222      2222111       3333
                       222111      2221111      11111222      111333
                       111111111   1111111111   22211111      222222
                                                11111111111   333111
                                                              11112222
                                                              22221111
                                                              111111222
                                                              111222111
                                                              222111111
                                                              111111111111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

The = 2 version is A003242 aerated.

The <= 1 version is A003242 ranked by A333489.

The version for parts instead of run-lengths is A078012, both A353428.

The version for partitions is A100405.

The > 1 version is A114901, ranked by A353427.

The <= 2 version is A128695, matching A335464.

A008466 counts compositions with some part > 2.

A011782 counts compositions.

A106356 counts compositions by number of adjacent equal parts.

A274174 counts compositions with equal parts contiguous.

A329738 counts uniform compositions, partitions A047966.

A329739 counts compositions with all distinct run-lengths.

Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.

Programs

Maple

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 17 2022

Mathematica

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

Extensions

a(21)-a(49) from Alois P. Heinz, May 17 2022