A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248

Offset

0,5

Comments

The Heinz numbers of these partitions are given by A325755.

Links

Table of n, a(n) for n=0..53.

Example

The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Mathematica

submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n], submultQ[Sort[Length/@Split[#]], #]&]], {n, 0, 30}]

Crossrefs

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

Sequence in context: A282249 A330953 A360308 * A304746 A304748 A323903

Adjacent sequences: A325699 A325700 A325701 * A325703 A325704 A325705

Keyword

nonn

Author

Gus Wiseman, May 18 2019

Status

approved