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In the ranked poset of integer partitions ordered by refinement, a(n) is the size of the lower ideal generated by the partition with Heinz number n.
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%I #10 Jul 25 2018 08:37:10

%S 1,1,2,1,3,2,5,1,3,3,7,2,11,5,5,1,15,3,22,3,8,7,30,2,6,11,4,5,42,5,56,

%T 1,11,15,11,3,77,22,17,3,101,8,135,7,7,30,176,2,14,6,23,11,231,4,15,5,

%U 33,42,297,5,385,56,11,1,23,11,490,15,45,11,627,3

%N In the ranked poset of integer partitions ordered by refinement, a(n) is the size of the lower ideal generated by the partition with Heinz number n.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The size of the corresponding upper ideal is A317141(n). Chains are A213427(n) and maximal chains are A002846(n).

%F a(prime(n)) = A000041(n).

%F a(x * y) <= a(x) * a(y).

%e The a(30) = 5 partitions are (321), (2211), (3111), (21111), (111111), with corresponding Heinz numbers: 30, 36, 40, 48, 64.

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[n]]]],{n,50}]

%Y Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273.

%Y Cf. A317141, A317142, A317143.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 04 2018