|
|
A358908
|
|
Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.
|
|
11
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
EXAMPLE
|
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))
|
|
MATHEMATICA
|
ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp], {comp, Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n], UnsameQ@@#&&GreaterEqual@@Length/@#&]], {n, 0, 10}]
|
|
PROG
|
(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
|
|
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.
A001970 counts multiset partitions of integer 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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|