cp's OEIS Frontend

A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

%I A358908 #9 Dec 31 2022 11:20:14
%S A358908 1,1,2,6,10,23,50,95,188,378,747,1414,2739,5179,9811,18562,34491,
%T A358908 64131,118607,218369,400196,731414,1328069,2406363,4346152,7819549,
%U A358908 14027500,25090582,44749372,79586074,141214698,249882141,441176493,777107137,1365801088,2395427040,4192702241
%N A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.
%H A358908 Andrew Howroyd, <a href="/A358908/b358908.txt">Table of n, a(n) for n = 0..200</a>
%e A358908 The a(1) = 1 through a(4) = 10 sequences:
%e A358908   ((1))  ((2))   ((3))      ((4))
%e A358908          ((11))  ((21))     ((22))
%e A358908                  ((111))    ((31))
%e A358908                  ((1)(2))   ((211))
%e A358908                  ((2)(1))   ((1111))
%e A358908                  ((11)(1))  ((1)(3))
%e A358908                             ((3)(1))
%e A358908                             ((11)(2))
%e A358908                             ((21)(1))
%e A358908                             ((111)(1))
%t A358908 ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t A358908 Table[Length[Select[ptnseq[n],UnsameQ@@#&&GreaterEqual@@Length/@#&]],{n,0,10}]
%o A358908 (PARI)
%o A358908 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358908 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 A358908 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 A358908 This is the distinct case of A055887 with weakly decreasing lengths.
%Y A358908 This is the distinct case is A141199.
%Y A358908 The case of distinct lengths also is A358836.
%Y A358908 This is the case of A358906 with weakly decreasing lengths.
%Y A358908 A000041 counts integer partitions, strict A000009.
%Y A358908 A001970 counts multiset partitions of integer partitions.
%Y A358908 A063834 counts twice-partitions.
%Y A358908 A358830 counts twice-partitions with distinct lengths.
%Y A358908 A358901 counts partitions with all distinct Omegas.
%Y A358908 A358912 counts sequences of partitions with distinct lengths.
%Y A358908 A358914 counts twice-partitions into distinct strict partitions.
%Y A358908 Cf. A000219, A261049, A271619, A296122, A358831, A358901, A358905, A358907.
%K A358908 nonn
%O A358908 0,3
%A A358908 _Gus Wiseman_, Dec 09 2022
%E A358908 Terms a(16) and beyond from _Andrew Howroyd_, Dec 31 2022