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Number of compositions of n with distinct first quotients.
11

%I #9 Mar 22 2021 15:01:45

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

%T 20231,33147

%N Number of compositions of n with distinct first quotients.

%C 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).

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.

%e The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).

%e The a(1) = 1 through a(5) = 13 compositions:

%e (1) (2) (3) (4) (5)

%e (1,1) (1,2) (1,3) (1,4)

%e (2,1) (2,2) (2,3)

%e (3,1) (3,2)

%e (1,1,2) (4,1)

%e (1,2,1) (1,1,3)

%e (2,1,1) (1,2,2)

%e (1,3,1)

%e (2,1,2)

%e (2,2,1)

%e (3,1,1)

%e (1,1,2,1)

%e (1,2,1,1)

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

%Y The version for differences instead of quotients is A325545.

%Y The version for equal first quotients is A342495.

%Y The unordered version is A342514, ranked by A342521.

%Y The strict unordered version is A342520.

%Y A000005 counts constant compositions.

%Y A000009 counts strictly increasing (or strictly decreasing) compositions.

%Y A000041 counts weakly increasing (or weakly decreasing) compositions.

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

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

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

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Mar 17 2021