proposed

approved

editing

proposed

Number of recursively totally strong compositions of n.

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

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

The total version case of partitions is A316496.

The case of partitions is (also) A316496.

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

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

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

Number of narrowly recursively strong compositions of n.

A sequence is narrowly recursively strong if either it is empty or a singleton equal to (narrow1), or its run-lengths are weakly decreasing (strong) and are themselves a narrowly recursively strong sequence (recursive).

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

tni[q_]:=Or[Length[q]<=={}, q=={1, }, And[GreaterEqual@@Length/@Split[q], tni[Length/@Split[q]]]];

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

allocated for Gus WisemanNumber of narrowly recursively 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

0,3

A sequence is narrowly recursively strong if either it is empty or a singleton (narrow), or its run-lengths are weakly decreasing (strong) and are themselves a narrowly recursively strong sequence (recursive).

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)

tni[q_]:=Or[Length[q]<=1, And[GreaterEqual@@Length/@Split[q], tni[Length/@Split[q]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tni]], {n, 0, 15}]

The total version is A316496.

The case of partitions is (also) A316496.

The case of reversed partitions is A332275.

The alternating version is A332338.

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

allocated

nonn,more

Gus Wiseman, Feb 11 2020

approved

editing

allocated for Gus Wiseman

allocated

approved