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Number of compositions of n into parts with distinct multiplicities and with exactly three parts.
5

%I #17 Nov 12 2020 22:19:28

%S 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,

%T 39,42,40,45,45,46,48,51,49,54,54,55,57,60,58,63,63,64,66,69,67,72,72,

%U 73,75,78,76,81,81,82,84,87,85,90,90,91,93,96,94,99,99

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

%H Alois P. Heinz, <a href="/A321773/b321773.txt">Table of n, a(n) for n = 3..1000</a>

%F Conjectures from _Colin Barker_, Dec 11 2018: (Start)

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

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

%F (End)

%e From _Gus Wiseman_, Nov 11 2020: (Start)

%e Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:

%e 111 112 113 114 115 116 117 118 119

%e 121 122 141 133 161 144 181 155

%e 211 131 222 151 224 171 226 191

%e 212 411 223 233 225 244 227

%e 221 232 242 252 262 272

%e 311 313 323 333 334 335

%e 322 332 414 343 344

%e 331 422 441 424 353

%e 511 611 522 433 434

%e 711 442 443

%e 622 515

%e 811 533

%e 551

%e 722

%e 911

%e (End)

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

%Y Column k=3 of A242887.

%Y A235451 counts 3-part compositions with distinct run-lengths

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

%Y A014311 intersected with A335488 ranks these compositions.

%Y A140106 is the unordered case, with Heinz numbers A285508.

%Y A261982 counts non-strict compositions of any length.

%Y A001523 counts unimodal compositions, with complement A115981.

%Y A007318 and A097805 count compositions by length.

%Y A032020 counts strict compositions.

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

%Y A242771 counts triples that are not strictly increasing.

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

%K nonn

%O 3,2

%A _Alois P. Heinz_, Nov 18 2018