A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830

Offset: 0

Author

Gus Wiseman, Dec 07 2022

Keywords

• nonn

Examples

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

The case of set partitions is A007837.

This is the case of A055887 with all distinct lengths.

For distinct sums instead of lengths we have A336342.

The case of twice-partitions is A358830.

The unordered version is A358836.

The version for constant instead of distinct lengths is A358905.

A000041 counts integer partitions, strict A000009.

A063834 counts twice-partitions.

A141199 counts sequences of partitions with weakly decreasing lengths.

Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.

Programs

Mathematica

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

PARI

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022

Extensions

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2022