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A353316 Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).

%I A353316 #10 May 16 2022 10:02:20
%S A353316 4,8,16,27,32,45,54,63,64,81,90,99,108,117,126,128,135,153,162,171,
%T A353316 180,189,198,207,216,234,243,252,256,261,270,279,297,306,324,333,342,
%U A353316 351,360,369,378,387,396,405,414,423,432,459,468,477,486,504,512,513,522
%N A353316 Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).
%C A353316 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 A353316 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 A353316 The terms together with their prime indices begin:
%e A353316     4: (1,1)
%e A353316     8: (1,1,1)
%e A353316    16: (1,1,1,1)
%e A353316    27: (2,2,2)
%e A353316    32: (1,1,1,1,1)
%e A353316    45: (3,2,2)
%e A353316    54: (2,2,2,1)
%e A353316    63: (4,2,2)
%e A353316    64: (1,1,1,1,1,1)
%e A353316    81: (2,2,2,2)
%e A353316    90: (3,2,2,1)
%e A353316    99: (5,2,2)
%e A353316   108: (2,2,2,1,1)
%e A353316   117: (6,2,2)
%e A353316   126: (4,2,2,1)
%e A353316   128: (1,1,1,1,1,1,1)
%e A353316 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 A353316 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353316 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A353316 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A353316 Select[Range[100],pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]==0&]
%Y A353316 These partitions are counted by A118199.
%Y A353316 Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.
%Y A353316 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A353316 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353316 A115720/A115994 count partitions by their Durfee square, rank stat A257990.
%Y A353316 A122111 represents partition conjugation using Heinz numbers.
%Y A353316 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A353316 A238394 counts reversed partitions without a fixed point, ranked by A352830.
%Y A353316 A238395 counts reversed partitions with a fixed point, ranked by A352872.
%Y A353316 A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
%Y A353316 A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
%Y A353316 Cf. A001222, A065770, A093641, A114088, A188674, A252464, A300788, A325163, A325169, A352831, A352828, A352829.
%K A353316 nonn
%O A353316 1,1
%A A353316 _Gus Wiseman_, May 15 2022