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A358832
Number of twice-partitions of n into partitions of distinct lengths and distinct sums.
4
1, 1, 2, 4, 7, 15, 25, 49, 79, 154, 248, 453, 748, 1305, 2125, 3702, 5931, 9990, 16415, 26844, 43246, 70947, 113653, 182314, 292897, 464614, 739640, 1169981, 1844511, 2888427, 4562850, 7079798, 11064182, 17158151, 26676385, 41075556, 63598025, 97420873, 150043132
OFFSET
0,3
COMMENTS
A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
EXAMPLE
The a(1) = 1 through a(5) = 15 twice-partitions:
(1) (2) (3) (4) (5)
(11) (21) (22) (32)
(111) (31) (41)
(11)(1) (211) (221)
(1111) (311)
(21)(1) (2111)
(111)(1) (11111)
(21)(2)
(22)(1)
(3)(11)
(31)(1)
(111)(2)
(211)(1)
(111)(11)
(1111)(1)
MATHEMATICA
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], UnsameQ@@Total/@#&&UnsameQ@@Length/@#&]], {n, 0, 10}]
PROG
(PARI)
seq(n)={ local(Cache=Map());
my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
my(F(m, r, b) = my(key=[m, r, b], z); if(!mapisdefined(Cache, key, &z),
z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m], k)); if(!bittest(b, k)&&c, c*self()(min(m-1, r-m), r-m, bitor(b, 1<<k)))));
mapput(Cache, key, z)); z);
vector(n+1, i, F(i-1, i-1, 0))
} \\ Andrew Howroyd, Dec 31 2022
CROSSREFS
This is the case of A271619 with distinct lengths.
These multiset partitions are ranked by A326535 /\ A326533.
This is the case of A358830 with distinct sums.
For constant instead of distinct lengths and sums we have A358833.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Sequence in context: A171276 A027167 A259090 * A232464 A264292 A259592
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2022
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022
STATUS
approved