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A350951
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Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.
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16
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0, 0, -2, 2, -2, 0, -4, 2, 0, 0, -4, 0, -6, -2, 0, 4, -6, 0, -8, 0, -2, -2, -8, 2, 2, -4, -2, -2, -10, 0, -10, 4, -2, -4, 0, 2, -12, -6, -4, 2, -12, -2, -14, -2, -2, -6, -14, 2, 0, 2, -4, -4, -16, 0, 0, 0, -6, -8, -16, 2, -18, -8, -4, 6, -2, -2, -18, -4, -6, 0
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OFFSET
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1,3
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COMMENTS
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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.
All terms are even.
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LINKS
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FORMULA
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a(A122111(n)) = -a(n), where A122111 represents partition conjugation.
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EXAMPLE
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The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 1 - 5 = -4.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Count[primeMS[n], _?OddQ]-Count[conj[primeMS[n]], _?OddQ], {n, 100}]
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CROSSREFS
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The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
The version comparing even with even conjugate parts is A350950.
There are four individual statistics:
There are five other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
A103919 counts partitions by number of odd parts.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A026424, A028260, A053738, A098123, A130780, A171966, A236559, A236914, A239241, A241638, A325700.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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