cp's OEIS Frontend

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

%I A336343 #12 Apr 17 2021 03:42:45
%S A336343 1,1,1,4,6,11,26,39,78,142,320,488,913,1558,2798,5865,9482,16742,
%T A336343 28474,50814,82800,172540,266093,472432,790824,1361460,2251665,
%U A336343 3844412,7205416,11370048,19483502,32416924,54367066,88708832,149179800,239738369,445689392
%N A336343 Number of ways to choose a strict partition of each part of a strict composition of n.
%C A336343 A strict composition of n (A032020) is a finite sequence of distinct positive integers summing to n.
%C A336343 Is there a simple generating function?
%H A336343 Andrew Howroyd, <a href="/A336343/b336343.txt">Table of n, a(n) for n = 0..1000</a>
%F A336343 G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000009(j)). - _Andrew Howroyd_, Apr 16 2021
%e A336343 The a(1) = 1 through a(5) = 11 ways:
%e A336343   (1)  (2)  (3)      (4)        (5)
%e A336343             (2,1)    (3,1)      (3,2)
%e A336343             (1),(2)  (1),(3)    (4,1)
%e A336343             (2),(1)  (3),(1)    (1),(4)
%e A336343                      (1),(2,1)  (2),(3)
%e A336343                      (2,1),(1)  (3),(2)
%e A336343                                 (4),(1)
%e A336343                                 (1),(3,1)
%e A336343                                 (2,1),(2)
%e A336343                                 (2),(2,1)
%e A336343                                 (3,1),(1)
%t A336343 strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
%t A336343 Table[Length[Join@@Table[Tuples[strptn/@ctn],{ctn,Join@@Permutations/@strptn[n]}]],{n,0,10}]
%o A336343 (PARI) \\ here Q(N) gives A000009 as a vector.
%o A336343 Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
%o A336343 seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+k] + O(x*x^n)))]} \\ _Andrew Howroyd_, Apr 16 2021
%Y A336343 Multiset partitions of partitions are A001970.
%Y A336343 Strict compositions are counted by A032020, A072574, and A072575.
%Y A336343 Splittings of strict partitions are A072706.
%Y A336343 Set partitions of strict partitions are A294617.
%Y A336343 Splittings of partitions with distinct sums are A336131.
%Y A336343 Cf. A008289, A011782, A304786, A318683, A318684, A319794, A323583, A336128, A336130, A336132, A336133.
%Y A336343 Partitions:
%Y A336343 - Partitions of each part of a partition are A063834.
%Y A336343 - Compositions of each part of a partition are A075900.
%Y A336343 - Strict partitions of each part of a partition are A270995.
%Y A336343 - Strict compositions of each part of a partition are A336141.
%Y A336343 Strict partitions:
%Y A336343 - Partitions of each part of a strict partition are A271619.
%Y A336343 - Compositions of each part of a strict partition are A304961.
%Y A336343 - Strict partitions of each part of a strict partition are A279785.
%Y A336343 - Strict compositions of each part of a strict partition are A336142.
%Y A336343 Compositions:
%Y A336343 - Partitions of each part of a composition are A055887.
%Y A336343 - Compositions of each part of a composition are A133494.
%Y A336343 - Strict partitions of each part of a composition are A304969.
%Y A336343 - Strict compositions of each part of a composition are A307068.
%Y A336343 Strict compositions:
%Y A336343 - Partitions of each part of a strict composition are A336342.
%Y A336343 - Compositions of each part of a strict composition are A336127.
%Y A336343 - Strict partitions of each part of a strict composition are A336343.
%Y A336343 - Strict compositions of each part of a strict composition are A336139.
%K A336343 nonn
%O A336343 0,4
%A A336343 _Gus Wiseman_, Jul 19 2020