A342493 Number of compositions of n with strictly increasing first quotients.

1, 1, 2, 3, 6, 8, 11, 16, 22, 28, 39, 49, 61, 77, 93, 114, 140, 169, 198, 233, 276, 321, 381, 439, 509, 591, 678, 774, 883, 1007, 1147, 1300, 1465, 1641, 1845, 2068, 2317, 2590, 2881, 3193, 3549, 3928, 4341, 4793, 5282, 5813, 6401, 7027, 7699, 8432, 9221, 10076

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.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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 unordered version is A342498, ranked by A342524.

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.

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A074206 counts ordered factorizations.

A167865 counts strict chains of divisors > 1 summing to n.

A274199 counts compositions with all adjacent parts x < 2y.

Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

Sequence in context: A185599 A211519 A329384 * A353903 A325547 A242340

Adjacent sequences: A342490 A342491 A342492 * A342494 A342495 A342496

Keyword

nonn

Author

Gus Wiseman, Mar 16 2021

Extensions

a(21)-a(51) from Alois P. Heinz, Mar 18 2021

Status

approved