A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

1, 3, 6, 4, 9, 9, 10, 12, 15, 13, 18, 18, 19, 21, 24, 22, 27, 27, 28, 30, 33, 31, 36, 36, 37, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99

Offset: 3

Author

Alois P. Heinz, Nov 18 2018

Keywords

• nonn
• changed

Examples

			From _Gus Wiseman_, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
  111   112   113   114   115   116   117   118   119
        121   122   141   133   161   144   181   155
        211   131   222   151   224   171   226   191
              212   411   223   233   225   244   227
              221         232   242   252   262   272
              311         313   323   333   334   335
                          322   332   414   343   344
                          331   422   441   424   353
                          511   611   522   433   434
                                      711   442   443
                                            622   515
                                            811   533
                                                  551
                                                  722
                                                  911
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A242887.

A235451 counts 3-part compositions with distinct run-lengths

A001399(n-6) counts 3-part compositions in the complement.

A014311 intersected with A335488 ranks these compositions.

A140106 is the unordered case, with Heinz numbers A285508.

A261982 counts non-strict compositions of any length.

A001523 counts unimodal compositions, with complement A115981.

A007318 and A097805 count compositions by length.

A032020 counts strict compositions.

A047967 counts non-strict partitions, with Heinz numbers A013929.

A242771 counts triples that are not strictly increasing.

Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.

Programs

Mathematica

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* Gus Wiseman, Nov 11 2020 *)

Formula

Conjectures from Colin Barker, Dec 11 2018: (Start)

G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).

a(n) = a(n-2) + a(n-3) - a(n-5) for n>7. (End)

Conjecture: a(n) = (3*n-k)/2 where k value has a cycle of 6 starting from n=3 of (7,6,3,10,3,6). - Bill McEachen, Aug 12 2025