A332274 Number of totally strong compositions of n.

1, 1, 2, 4, 7, 11, 22, 33, 56, 93, 162, 264, 454, 765, 1307, 2237, 3849, 6611, 11472, 19831, 34446, 59865, 104293, 181561, 316924

Offset: 0

Author

Gus Wiseman, Feb 11 2020

Keywords

• nonn
• more

Comments

A sequence is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong sequence.

A composition of n is a finite sequence of positive integers with sum n.

Also the number of totally co-strong compositions of n.

Examples

			The a(1) = 1 through a(5) = 11 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (121)   (41)
                    (211)   (122)
                    (1111)  (131)
                            (212)
                            (311)
                            (2111)
                            (11111)
		

Crossrefs

The case of partitions is A316496.

The co-strong case is A332274 (this sequence).

The case of reversed partitions is A332275.

The alternating version is A332338.

Cf. A100883, A107429, A317245, A317256, A317491, A329744, A332272, A332279, A332289, A332292, A332336, A332337, A332339, A332340.

Programs

Mathematica

tni[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],tni[Length/@Split[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tni]],{n,0,15}]