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%I #11 Dec 31 2022 20:30:13
%S 1,1,2,4,9,15,31,53,105,178,330,555,1024,1693,2991,5014,8651,14242,
%T 24477,39864,67078,109499,181311,292764,483775,774414,1260016,2016427,
%U 3254327,5162407,8285796,13074804,20812682,32733603,51717463,80904644,127305773,198134675,309677802
%N Number of twice-partitions of n into partitions with all different lengths.
%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%e The a(1) = 1 through a(5) = 15 twice-partitions:
%e (1) (2) (3) (4) (5)
%e (11) (21) (22) (32)
%e (111) (31) (41)
%e (11)(1) (211) (221)
%e (1111) (311)
%e (11)(2) (2111)
%e (2)(11) (11111)
%e (21)(1) (21)(2)
%e (111)(1) (22)(1)
%e (3)(11)
%e (31)(1)
%e (111)(2)
%e (211)(1)
%e (111)(11)
%e (1111)(1)
%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t Table[Length[Select[twiptn[n],UnsameQ@@Length/@#&]],{n,0,10}]
%o (PARI)
%o seq(n)={ local(Cache=Map());
%o my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
%o my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
%o 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,r-m), r-m, bitor(b, 1<<k)))));
%o mapput(Cache, key, z)); z);
%o vector(n+1, i, F(i-1, i-1, 0))
%o } \\ _Andrew Howroyd_, Dec 31 2022
%Y The version for set partitions is A007837.
%Y For sums instead of lengths we have A271619.
%Y For constant instead of distinct lengths we have A306319.
%Y The case of distinct sums also is A358832.
%Y The version for multiset partitions of integer partitions is A358836.
%Y A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A273873 counts strict trees.
%Y Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A336342, A358334, A358831.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 03 2022
%E Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022
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