A355383 - OEIS

%I #9 Jul 03 2022 23:56:28

%S 1,1,2,3,6,10,16,26,42,64,100,150,224,330,482,697,999,1418,1996,2794,

%T 3879,5355,7343,10018,13583,18338,24618,32917,43790,58043,76591,

%U 100716,131906,172194,223966,290423,375318,483668,621368,796138,1017146

%N 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 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 The a(0) = 1 through a(5) = 10 pairs:

%e   ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)

%e                 (11)(1)  (21)(21)  (31)(31)   (41)(41)

%e                          (111)(1)  (22)(2)    (32)(32)

%e                                    (211)(11)  (311)(11)

%e                                    (211)(21)  (311)(31)

%e                                    (1111)(1)  (221)(21)

%e                                               (221)(22)

%e                                               (2111)(11)

%e                                               (2111)(21)

%e                                               (11111)(1)

%t Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

%Y With multiplicity we have A339006.

%Y The version for compositions is A355384.

%Y The homogeneous version w/o containment is A355385, compositions A355388.

%Y A001970 counts multiset partitions of partitions.

%Y A063834 counts partitions of each part of a partition.

%Y Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.

%Y Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 02 2022