A325702 - OEIS

%I #8 Jul 05 2019 18:32:19

%S 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,

%T 195,233,285,348,427,506,613,733,873,1063,1263,1503,1802,2131,2537,

%U 3005,3565,4171,4922,5820,6775,8001,9333,10860,12739,14840,17206,20029,23248

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

%C The Heinz numbers of these partitions are given by A325755.

%e 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).

%e The a(1) = 1 through a(11) = 7 partitions:

%e   (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)

%e        (211)         (3111)          (32111)  (5211)   (3331)    (33221)

%e                                      (41111)  (32211)  (6211)    (52211)

%e                                                        (42211)   (53111)

%e                                                        (43111)   (322211)

%e                                                        (322111)  (332111)

%e                                                        (421111)  (431111)

%e                                                        (511111)

%t submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]

%t Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

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

%K nonn

%O 0,5

%A _Gus Wiseman_, May 18 2019