OFFSET
0,3
COMMENTS
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 (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7).
The a(1) = 1 through a(7) = 16 compositions:
(1) (2) (3) (4) (5) (6) (7)
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5)
(3,1) (3,2) (3,3) (3,4)
(1,1,2) (4,1) (4,2) (4,3)
(2,1,1) (1,1,3) (5,1) (5,2)
(2,1,2) (1,1,4) (6,1)
(3,1,1) (2,1,3) (1,1,5)
(3,1,2) (2,1,4)
(4,1,1) (2,2,3)
(2,1,1,2) (3,1,3)
(3,2,2)
(4,1,2)
(5,1,1)
(2,1,1,3)
(3,1,1,2)
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..55); # Alois P. Heinz, Mar 25 2021
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@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, 55] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
CROSSREFS
The version for differences instead of quotients is A325547.
The weakly increasing version is A342492.
The strictly decreasing version is A342494.
The strict unordered version is A342517.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A274199 counts compositions with all adjacent parts x < 2y.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 16 2021
EXTENSIONS
a(21)-a(51) from Alois P. Heinz, Mar 18 2021
STATUS
approved