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A353317 Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).

%I A353317 #9 May 16 2022 10:02:25
%S A353317 2,9,15,18,21,30,33,36,39,42,51,57,60,66,69,72,78,84,87,93,102,111,
%T A353317 114,120,123,125,129,132,138,141,144,156,159,168,174,175,177,183,186,
%U A353317 201,204,213,219,222,228,237,240,245,246,249,250,258,264,267,275,276
%N A353317 Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).
%C A353317 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 A353317 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 A353317 The terms and their prime indices begin:
%e A353317     2: (1)
%e A353317     9: (2,2)
%e A353317    15: (3,2)
%e A353317    18: (2,2,1)
%e A353317    21: (4,2)
%e A353317    30: (3,2,1)
%e A353317    33: (5,2)
%e A353317    36: (2,2,1,1)
%e A353317    39: (6,2)
%e A353317    42: (4,2,1)
%e A353317    51: (7,2)
%e A353317    57: (8,2)
%e A353317    60: (3,2,1,1)
%e A353317    66: (5,2,1)
%e A353317    69: (9,2)
%e A353317    72: (2,2,1,1,1)
%e A353317    78: (6,2,1)
%e A353317    84: (4,2,1,1)
%e A353317 For example, the partition (2,2,1,1) with Heinz number 36 has a fixed point at the second position, as does its conjugate (4,2), so 36 is in the sequence.
%t A353317 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353317 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A353317 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A353317 Select[Range[100],pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]>0&]
%Y A353317 These partitions are counted by A188674.
%Y A353317 Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.
%Y A353317 The strict case is A352829.
%Y A353317 Fixed point but no conjugate fixed point: A353316, counted by A118199.
%Y A353317 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A353317 A002467 counts permutations with a fixed point, complement A000166.
%Y A353317 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353317 A115720/A115994 count partitions by their Durfee square, rank stat A257990.
%Y A353317 A122111 represents partition conjugation using Heinz numbers.
%Y A353317 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A353317 A238394 counts reversed partitions without a fixed point, ranked by A352830.
%Y A353317 A238395 counts reversed partitions with a fixed point, ranked by A352872.
%Y A353317 A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
%Y A353317 A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
%Y A353317 Cf. A001222, A065770, A093641, A252464, A325039, A325163, A325169, A352828, A352831, A352832, A352833.
%K A353317 nonn
%O A353317 1,1
%A A353317 _Gus Wiseman_, May 15 2022