A342494 Number of compositions of n with strictly decreasing first quotients.

1, 1, 2, 3, 5, 8, 12, 15, 21, 30, 39, 50, 65, 82, 103, 129, 160, 196, 240, 293, 352, 422, 500, 593, 706, 832, 974, 1138, 1324, 1534, 1783, 2054, 2362, 2712, 3108, 3552, 4051, 4606, 5232, 5935, 6713, 7573, 8536, 9597, 10773, 12085, 13534, 15119, 16874, 18809

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

Table of n, a(n) for n=0..49.

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 (1,2,3,4,2) has first quotients (2,3/2,4/3,1/2) so is counted under a(12).
The a(1) = 1 through a(6) = 12 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)
                     (3,1)    (3,2)    (3,3)
                     (1,2,1)  (4,1)    (4,2)
                              (1,2,2)  (5,1)
                              (1,3,1)  (1,2,3)
                              (2,2,1)  (1,3,2)
                                       (1,4,1)
                                       (2,3,1)
                                       (3,2,1)
                                       (1,2,2,1)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Greater@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}]

Crossrefs

The weakly decreasing version is A069916.

The version for differences instead of quotients is A325548.

The strictly increasing version is A342493.

The unordered version is A342499, ranked by A342525.

The strict unordered version is A342518.

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: A308075 A211181 A028766 * A022487 A305377 A062489

Adjacent sequences: A342491 A342492 A342493 * A342495 A342496 A342497

Keyword

nonn

Author

Gus Wiseman, Mar 17 2021

Extensions

a(21)-a(49) from Alois P. Heinz, Mar 18 2021

Status

approved