A353316
Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).
Original entry on oeis.org
4, 8, 16, 27, 32, 45, 54, 63, 64, 81, 90, 99, 108, 117, 126, 128, 135, 153, 162, 171, 180, 189, 198, 207, 216, 234, 243, 252, 256, 261, 270, 279, 297, 306, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 459, 468, 477, 486, 504, 512, 513, 522
Offset: 1
The terms together with their prime indices begin:
4: (1,1)
8: (1,1,1)
16: (1,1,1,1)
27: (2,2,2)
32: (1,1,1,1,1)
45: (3,2,2)
54: (2,2,2,1)
63: (4,2,2)
64: (1,1,1,1,1,1)
81: (2,2,2,2)
90: (3,2,2,1)
99: (5,2,2)
108: (2,2,2,1,1)
117: (6,2,2)
126: (4,2,2,1)
128: (1,1,1,1,1,1,1)
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.
These partitions are counted by
A118199.
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,
A114088,
A188674,
A252464,
A300788,
A325163,
A325169,
A352831,
A352828,
A352829.
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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&]
A374782
Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.
Original entry on oeis.org
1, 0, 2, 3, 4, 5, 8, 11, 17, 23, 33, 43, 60, 77, 104, 134, 177, 226, 295, 373, 480, 604, 766, 957, 1204, 1492, 1860, 2294, 2836, 3477, 4273, 5209, 6362, 7721, 9375, 11326, 13687, 16460, 19799, 23720, 28406, 33901, 40443, 48092, 57159, 67747, 80237, 94799
Offset: 0
a(0) = 1: the empty partition.
a(2) = 2: 2, 11.
a(3) = 3: 3, 21, 111.
a(4) = 4: 4, 31, 211, 1111.
a(5) = 5: 5, 41, 311, 2111, 11111.
a(6) = 8: 6, 33, 51, 222, 411, 3111, 21111, 111111.
a(7) = 11: 7, 43, 61, 322, 331, 511, 2221, 4111, 31111, 211111, 1111111.
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b:= proc(n, i, p) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1, p)+add(`if`(i=p+j, 0,
b(n-i*j, min(n-i*j, i-1), p+j)), j=1..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..47);
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