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A325702
Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).
25
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 18 2019
STATUS
approved