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

2, 9, 15, 18, 21, 30, 33, 36, 39, 42, 51, 57, 60, 66, 69, 72, 78, 84, 87, 93, 102, 111, 114, 120, 123, 125, 129, 132, 138, 141, 144, 156, 159, 168, 174, 175, 177, 183, 186, 201, 204, 213, 219, 222, 228, 237, 240, 245, 246, 249, 250, 258, 264, 267, 275, 276

Offset: 1

Author

Gus Wiseman, May 15 2022

Keywords

• nonn

Comments

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

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.

Examples

			The terms and their prime indices begin:
    2: (1)
    9: (2,2)
   15: (3,2)
   18: (2,2,1)
   21: (4,2)
   30: (3,2,1)
   33: (5,2)
   36: (2,2,1,1)
   39: (6,2)
   42: (4,2,1)
   51: (7,2)
   57: (8,2)
   60: (3,2,1,1)
   66: (5,2,1)
   69: (9,2)
   72: (2,2,1,1,1)
   78: (6,2,1)
   84: (4,2,1,1)
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.
		

Crossrefs

These partitions are counted by A188674.

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

The strict case is A352829.

Fixed point but no conjugate fixed point: A353316, counted by A118199.

A000700 counts self-conjugate partitions, ranked by A088902.

A002467 counts permutations with a fixed point, complement A000166.

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

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

A122111 represents partition conjugation using Heinz numbers.

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

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

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

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

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

Cf. A001222, A065770, A093641, A252464, A325039, A325163, A325169, A352828, A352831, A352832, A352833.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]>0&]