cp's OEIS Frontend

A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

%I A355385 #10 Dec 31 2022 20:30:03
%S A355385 1,1,2,3,7,12,25,43,81,141,243,409,699,1132,1844,2995,4744,7408,11655,
%T A355385 17839,27509,41546,62879,93537,139974,205547,302714,440097,640968,
%U A355385 921774,1327538,1891548,2696635,3809860,5380257,7540778,10561566,14687109,20408170,28183998,38882009
%N A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.
%C A355385 Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?
%H A355385 Andrew Howroyd, <a href="/A355385/b355385.txt">Table of n, a(n) for n = 0..1000</a>
%F A355385 a(n) = Sum_{j >= 1} A116608(n,j) * A008284(n,j) for n > 0. - _Andrew Howroyd_, Dec 31 2022
%e A355385 The a(0) = 1 through a(5) = 10 pairs:
%e A355385   ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
%e A355385                 (11)(2)  (21)(21)  (31)(31)   (41)(41)
%e A355385                          (111)(3)  (31)(22)   (41)(32)
%e A355385                                    (22)(4)    (32)(41)
%e A355385                                    (211)(31)  (32)(32)
%e A355385                                    (211)(22)  (311)(41)
%e A355385                                    (1111)(4)  (311)(32)
%e A355385                                               (221)(41)
%e A355385                                               (221)(32)
%e A355385                                               (2111)(41)
%e A355385                                               (2111)(32)
%e A355385                                               (11111)(5)
%t A355385 Table[Length[Select[Tuples[IntegerPartitions[n],2],Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,15}]
%o A355385 (PARI) \\ P gives A008284 and R gives A116608 as g.f.'s.
%o A355385 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A355385 R(n,y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
%o A355385 seq(n) = {my(g=Vec(P(n,y)), h=Vec(R(n,y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p,j)*polcoef(q,j)))} \\ _Andrew Howroyd_, Dec 31 2022
%Y A355385 The inhomogeneous version with containment and multiplicity is A339006.
%Y A355385 The inhomogeneous version with containment is A355383.
%Y A355385 The inhomogeneous version with containment for compositions is A355384.
%Y A355385 The version for compositions is A355388.
%Y A355385 A001970 counts multiset partitions of partitions.
%Y A355385 A063834 counts partitions of each part of a partition.
%Y A355385 A323583 counts splittings of partitions.
%Y A355385 Cf. A000009, A008284, A022811, A032020, A070933, A072706, A116608, A279787, A319910, A336135.
%K A355385 nonn
%O A355385 0,3
%A A355385 _Gus Wiseman_, Jul 02 2022
%E A355385 Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022