This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355383 #9 Jul 03 2022 23:56:28
%S A355383 1,1,2,3,6,10,16,26,42,64,100,150,224,330,482,697,999,1418,1996,2794,
%T A355383 3879,5355,7343,10018,13583,18338,24618,32917,43790,58043,76591,
%U A355383 100716,131906,172194,223966,290423,375318,483668,621368,796138,1017146
%N A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.
%C A355383 If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.
%e A355383 The a(0) = 1 through a(5) = 10 pairs:
%e A355383 ()() (1)(1) (2)(2) (3)(3) (4)(4) (5)(5)
%e A355383 (11)(1) (21)(21) (31)(31) (41)(41)
%e A355383 (111)(1) (22)(2) (32)(32)
%e A355383 (211)(11) (311)(11)
%e A355383 (211)(21) (311)(31)
%e A355383 (1111)(1) (221)(21)
%e A355383 (221)(22)
%e A355383 (2111)(11)
%e A355383 (2111)(21)
%e A355383 (11111)(1)
%t A355383 Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]
%Y A355383 With multiplicity we have A339006.
%Y A355383 The version for compositions is A355384.
%Y A355383 The homogeneous version w/o containment is A355385, compositions A355388.
%Y A355383 A001970 counts multiset partitions of partitions.
%Y A355383 A063834 counts partitions of each part of a partition.
%Y A355383 Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
%Y A355383 Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.
%K A355383 nonn
%O A355383 0,3
%A A355383 _Gus Wiseman_, Jul 02 2022