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, 53973, 87254, 140861, 227535, 368050, 589706, 940999, 1497912, 2378260, 3774297, 5964712, 9416411, 14822087, 23244440, 36420756

Offset: 0

Author

Gus Wiseman, Mar 17 2021

Keywords

• nonn
• more

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).

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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

Extensions

a(21)-a(35) from Alois P. Heinz, Jan 16 2025