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

%I A358912 #12 Dec 31 2022 11:20:20
%S A358912 1,1,2,5,11,23,49,103,214,434,874,1738,3443,6765,13193,25512,48957,
%T A358912 93267,176595,332550,622957,1161230,2153710,3974809,7299707,13343290,
%U A358912 24280924,43999100,79412942,142792535,255826836,456735456,812627069,1440971069,2546729830
%N A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.
%H A358912 Andrew Howroyd, <a href="/A358912/b358912.txt">Table of n, a(n) for n = 0..1000</a>
%e A358912 The a(1) = 1 through a(4) = 11 sequences:
%e A358912   (1)  (2)   (3)      (4)
%e A358912        (11)  (21)     (22)
%e A358912              (111)    (31)
%e A358912              (1)(11)  (211)
%e A358912              (11)(1)  (1111)
%e A358912                       (11)(2)
%e A358912                       (1)(21)
%e A358912                       (2)(11)
%e A358912                       (21)(1)
%e A358912                       (1)(111)
%e A358912                       (111)(1)
%t A358912 ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t A358912 Table[Length[Select[ptnseq[n],UnsameQ@@Length/@#&]],{n,0,10}]
%o A358912 (PARI)
%o A358912 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358912 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
%Y A358912 The case of set partitions is A007837.
%Y A358912 This is the case of A055887 with all distinct lengths.
%Y A358912 For distinct sums instead of lengths we have A336342.
%Y A358912 The case of twice-partitions is A358830.
%Y A358912 The unordered version is A358836.
%Y A358912 The version for constant instead of distinct lengths is A358905.
%Y A358912 A000041 counts integer partitions, strict A000009.
%Y A358912 A063834 counts twice-partitions.
%Y A358912 A141199 counts sequences of partitions with weakly decreasing lengths.
%Y A358912 Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.
%K A358912 nonn
%O A358912 0,3
%A A358912 _Gus Wiseman_, Dec 07 2022
%E A358912 Terms a(16) and beyond from _Andrew Howroyd_, Dec 30 2022