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Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).
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%I #10 May 16 2022 10:02:20

%S 4,8,16,27,32,45,54,63,64,81,90,99,108,117,126,128,135,153,162,171,

%T 180,189,198,207,216,234,243,252,256,261,270,279,297,306,324,333,342,

%U 351,360,369,378,387,396,405,414,423,432,459,468,477,486,504,512,513,522

%N Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).

%C A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%e The terms together with their prime indices begin:

%e 4: (1,1)

%e 8: (1,1,1)

%e 16: (1,1,1,1)

%e 27: (2,2,2)

%e 32: (1,1,1,1,1)

%e 45: (3,2,2)

%e 54: (2,2,2,1)

%e 63: (4,2,2)

%e 64: (1,1,1,1,1,1)

%e 81: (2,2,2,2)

%e 90: (3,2,2,1)

%e 99: (5,2,2)

%e 108: (2,2,2,1,1)

%e 117: (6,2,2)

%e 126: (4,2,2,1)

%e 128: (1,1,1,1,1,1,1)

%e For example, the partition (3,2,2,1) with Heinz number 90 has a fixed point at the second position, but its conjugate (4,3,1) has no fixed points, so 90 is in the sequence.

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

%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Select[Range[100],pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]==0&]

%Y These partitions are counted by A118199.

%Y Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A115720/A115994 count partitions by their Durfee square, rank stat A257990.

%Y A122111 represents partition conjugation using Heinz numbers.

%Y A238352 counts reversed partitions by fixed points, rank statistic A352822.

%Y A238394 counts reversed partitions without a fixed point, ranked by A352830.

%Y A238395 counts reversed partitions with a fixed point, ranked by A352872.

%Y A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).

%Y A352827 ranks partitions with a fixed point, counted by A001522 (unproved).

%Y Cf. A001222, A065770, A093641, A114088, A188674, A252464, A300788, A325163, A325169, A352831, A352828, A352829.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 15 2022