A342529 Number of compositions of n with distinct first quotients.

1, 1, 2, 3, 7, 13, 19, 36, 67, 114, 197, 322, 564, 976, 1614, 2729, 4444, 7364, 12357, 20231, 33147

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..20.

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

Example

The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)    (3)    (4)      (5)
       (1,1)  (1,2)  (1,3)    (1,4)
              (2,1)  (2,2)    (2,3)
                     (3,1)    (3,2)
                     (1,1,2)  (4,1)
                     (1,2,1)  (1,1,3)
                     (2,1,1)  (1,2,2)
                              (1,3,1)
                              (2,1,2)
                              (2,2,1)
                              (3,1,1)
                              (1,1,2,1)
                              (1,2,1,1)

Mathematica

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

Crossrefs

The version for differences instead of quotients is A325545.

The version for equal first quotients is A342495.

The unordered version is A342514, ranked by A342521.

The strict unordered version is A342520.

A000005 counts constant compositions.

A000009 counts strictly increasing (or strictly decreasing) compositions.

A000041 counts weakly increasing (or weakly decreasing) compositions.

A001055 counts factorizations (strict: A045778, ordered: A074206).

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

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

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

Sequence in context: A013645 A130272 A331948 * A172238 A319496 A038940

Adjacent sequences: A342526 A342527 A342528 * A342530 A342531 A342532

Keyword

nonn,more

Author

Gus Wiseman, Mar 17 2021

Status

approved