%I #9 Dec 31 2022 11:20:14

%S 1,1,2,6,10,23,50,95,188,378,747,1414,2739,5179,9811,18562,34491,

%T 64131,118607,218369,400196,731414,1328069,2406363,4346152,7819549,

%U 14027500,25090582,44749372,79586074,141214698,249882141,441176493,777107137,1365801088,2395427040,4192702241

%N Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

%H Andrew Howroyd, <a href="/A358908/b358908.txt">Table of n, a(n) for n = 0..200</a>

%e The a(1) = 1 through a(4) = 10 sequences:

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

%e ((11)) ((21)) ((22))

%e ((111)) ((31))

%e ((1)(2)) ((211))

%e ((2)(1)) ((1111))

%e ((11)(1)) ((1)(3))

%e ((3)(1))

%e ((11)(2))

%e ((21)(1))

%e ((111)(1))

%t ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];

%t Table[Length[Select[ptnseq[n],UnsameQ@@#&&GreaterEqual@@Length/@#&]],{n,0,10}]

%o (PARI)

%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}

%o R(n,v) = {[subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, #v, (1 + y*x^k + O(x*x^n))^v[k] ))]}

%o seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, Ser(R(n, Vec(polcoef(g, k, y), -n))) ))} \\ _Andrew Howroyd_, Dec 31 2022

%Y This is the distinct case of A055887 with weakly decreasing lengths.

%Y This is the distinct case is A141199.

%Y The case of distinct lengths also is A358836.

%Y This is the case of A358906 with weakly decreasing lengths.

%Y A000041 counts integer partitions, strict A000009.

%Y A001970 counts multiset partitions of integer partitions.

%Y A063834 counts twice-partitions.

%Y A358830 counts twice-partitions with distinct lengths.

%Y A358901 counts partitions with all distinct Omegas.

%Y A358912 counts sequences of partitions with distinct lengths.

%Y A358914 counts twice-partitions into distinct strict partitions.

%Y Cf. A000219, A261049, A271619, A296122, A358831, A358901, A358905, A358907.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 09 2022

%E Terms a(16) and beyond from _Andrew Howroyd_, Dec 31 2022