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

1, 1, 2, 6, 10, 23, 50, 95, 188, 378, 747, 1414, 2739, 5179, 9811, 18562, 34491, 64131, 118607, 218369, 400196, 731414, 1328069, 2406363, 4346152, 7819549, 14027500, 25090582, 44749372, 79586074, 141214698, 249882141, 441176493, 777107137, 1365801088, 2395427040, 4192702241

Offset: 0

Author

Gus Wiseman, Dec 09 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(4) = 10 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((11)(1))  ((1)(3))
                            ((3)(1))
                            ((11)(2))
                            ((21)(1))
                            ((111)(1))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

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

This is the distinct case is A141199.

The case of distinct lengths also is A358836.

This is the case of A358906 with weakly decreasing lengths.

A000041 counts integer partitions, strict A000009.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions.

A358830 counts twice-partitions with distinct lengths.

A358901 counts partitions with all distinct Omegas.

A358912 counts sequences of partitions with distinct lengths.

A358914 counts twice-partitions into distinct strict partitions.

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

Programs

Mathematica

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@#&&GreaterEqual@@Length/@#&]],{n,0,10}]

PARI

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

Extensions

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022