OFFSET
0,3
COMMENTS
Also called log-concave-up compositions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
EXAMPLE
The composition (4,2,1,2,3) has first quotients (1/2,1/2,2,3/2) so is not counted under a(12), even though the first differences (-2,-1,1,1) are weakly increasing.
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)
(1,1,1) (3,1) (3,2) (3,3)
(1,1,2) (4,1) (4,2)
(2,1,1) (1,1,3) (5,1)
(1,1,1,1) (2,1,2) (1,1,4)
(3,1,1) (2,1,3)
(1,1,1,2) (2,2,2)
(2,1,1,1) (3,1,2)
(1,1,1,1,1) (4,1,1)
(1,1,1,3)
(2,1,1,2)
(3,1,1,1)
(1,1,1,1,2)
(2,1,1,1,1)
(1,1,1,1,1,1)
MAPLE
b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
`if`(q=0 or q>=l/j, b(n-j, l/j, j), 0), j=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 25 2021
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], LessEqual@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
If[q == 0 || q >= l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
CROSSREFS
The weakly decreasing version is A069916.
The version for differences instead of quotients is A325546.
The strictly increasing version is A342493.
The strict unordered version is A342516.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A002843 counts compositions with all adjacent parts x <= 2y.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 16 2021
EXTENSIONS
a(21)-a(47) from Alois P. Heinz, Mar 25 2021
STATUS
approved