cp's OEIS Frontend

A336342 Number of ways to choose a partition of each part of a strict composition of n.

%I A336342 #14 Apr 17 2021 03:42:26
%S A336342 1,1,2,7,11,29,81,155,312,708,1950,3384,7729,14929,32407,81708,151429,
%T A336342 305899,623713,1234736,2463743,6208978,10732222,22487671,43000345,
%U A336342 86573952,160595426,324990308,744946690,1336552491,2629260284,5050032692,9681365777
%N A336342 Number of ways to choose a partition of each part of a strict composition of n.
%C A336342 A strict composition of n is a finite sequence of distinct positive integers summing to n.
%C A336342 Is there a simple generating function?
%H A336342 Andrew Howroyd, <a href="/A336342/b336342.txt">Table of n, a(n) for n = 0..1000</a>
%F A336342 G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000041(j)). - _Andrew Howroyd_, Apr 16 2021
%e A336342 The a(1) = 1 through a(4) = 11 ways:
%e A336342   (1)  (2)    (3)        (4)
%e A336342        (1,1)  (2,1)      (2,2)
%e A336342               (1,1,1)    (3,1)
%e A336342               (1),(2)    (1),(3)
%e A336342               (2),(1)    (2,1,1)
%e A336342               (1),(1,1)  (3),(1)
%e A336342               (1,1),(1)  (1,1,1,1)
%e A336342                          (1),(2,1)
%e A336342                          (2,1),(1)
%e A336342                          (1),(1,1,1)
%e A336342                          (1,1,1),(1)
%t A336342 Table[Length[Join@@Table[Tuples[IntegerPartitions/@ctn],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,10}]
%o A336342 (PARI) seq(n)={[subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*numbpart(k) + O(x*x^n)))]} \\ _Andrew Howroyd_, Apr 16 2021
%Y A336342 Multiset partitions of partitions are A001970.
%Y A336342 Strict compositions are counted by A032020, A072574, and A072575.
%Y A336342 Splittings of partitions are A323583.
%Y A336342 Splittings of partitions with distinct sums are A336131.
%Y A336342 Cf. A008289, A011782, A304786, A316245, A317715, A336128, A336132, A336134, A336135.
%Y A336342 Partitions:
%Y A336342 - Partitions of each part of a partition are A063834.
%Y A336342 - Compositions of each part of a partition are A075900.
%Y A336342 - Strict partitions of each part of a partition are A270995.
%Y A336342 - Strict compositions of each part of a partition are A336141.
%Y A336342 Strict partitions:
%Y A336342 - Partitions of each part of a strict partition are A271619.
%Y A336342 - Compositions of each part of a strict partition are A304961.
%Y A336342 - Strict partitions of each part of a strict partition are A279785.
%Y A336342 - Strict compositions of each part of a strict partition are A336142.
%Y A336342 Compositions:
%Y A336342 - Partitions of each part of a composition are A055887.
%Y A336342 - Compositions of each part of a composition are A133494.
%Y A336342 - Strict partitions of each part of a composition are A304969.
%Y A336342 - Strict compositions of each part of a composition are A307068.
%Y A336342 Strict compositions:
%Y A336342 - Partitions of each part of a strict composition are A336342.
%Y A336342 - Compositions of each part of a strict composition are A336127.
%Y A336342 - Strict partitions of each part of a strict composition are A336343.
%Y A336342 - Strict compositions of each part of a strict composition are A336139.
%K A336342 nonn
%O A336342 0,3
%A A336342 _Gus Wiseman_, Jul 18 2020