cp's OEIS Frontend

A358906 Number of finite sequences of distinct integer partitions with total sum n.

%I A358906 #22 Feb 13 2024 19:42:43
%S A358906 1,1,2,7,13,35,87,191,470,1080,2532,5778,13569,30715,69583,160386,
%T A358906 360709,814597,1824055,4102430,9158405,20378692,45215496,100055269,
%U A358906 221388993,486872610,1069846372,2343798452,5127889666,11186214519,24351106180,52896439646
%N A358906 Number of finite sequences of distinct integer partitions with total sum n.
%H A358906 Alois P. Heinz, <a href="/A358906/b358906.txt">Table of n, a(n) for n = 0..1000</a>
%F A358906 a(n) = Sum_{k} A330463(n,k) * k!.
%e A358906 The a(1) = 1 through a(4) = 13 sequences:
%e A358906   ((1))  ((2))   ((3))      ((4))
%e A358906          ((11))  ((21))     ((22))
%e A358906                  ((111))    ((31))
%e A358906                  ((1)(2))   ((211))
%e A358906                  ((2)(1))   ((1111))
%e A358906                  ((1)(11))  ((1)(3))
%e A358906                  ((11)(1))  ((3)(1))
%e A358906                             ((11)(2))
%e A358906                             ((1)(21))
%e A358906                             ((2)(11))
%e A358906                             ((21)(1))
%e A358906                             ((1)(111))
%e A358906                             ((111)(1))
%p A358906 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p A358906       binomial(combinat[numbpart](i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
%p A358906     end:
%p A358906 a:= n-> b(n$2, 0):
%p A358906 seq(a(n), n=0..32);  # _Alois P. Heinz_, Feb 13 2024
%t A358906 ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t A358906 Table[Length[Select[ptnseq[n],UnsameQ@@#&]],{n,0,10}]
%Y A358906 This is the case of A055887 with distinct partitions.
%Y A358906 The unordered version is A261049.
%Y A358906 The case of twice-partitions is A296122.
%Y A358906 The case of distinct sums is A336342, constant sums A279787.
%Y A358906 The version for sequences of compositions is A358907.
%Y A358906 The case of weakly decreasing lengths is A358908.
%Y A358906 The case of distinct lengths is A358912.
%Y A358906 The version for strict partitions is A358913, distinct case of A304969.
%Y A358906 A001970 counts multiset partitions of integer partitions.
%Y A358906 A063834 counts twice-partitions.
%Y A358906 A358830 counts twice-partitions with distinct lengths.
%Y A358906 A358901 counts partitions with all distinct Omegas.
%Y A358906 Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.
%K A358906 nonn
%O A358906 0,3
%A A358906 _Gus Wiseman_, Dec 07 2022