A358830 - OEIS

%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