A350951 Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.

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

Offset: 1

Author

Gus Wiseman, Mar 14 2022

Keywords

• sign

Comments

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.

Examples

			The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 1 - 5 = -4.
		

Crossrefs

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.

Positions of 0's are A350944, even rank case A345196, counted by A277103.

The version comparing even with even conjugate parts is A350950.

There are four individual statistics:

- A257991 counts odd parts, conjugate A344616.

- A257992 counts even parts, conjugate A350847.

There are five other possible pairings of statistics:

- A325698: # of even parts = # of odd parts, counted by A045931.

- 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:

- A350946, counted by A351977.

- A350949, counted by A351976.

- A351980, counted by A351981.

The case of all four statistics equal is A350947, counted by A351978.

A056239 adds up prime indices, counted by A001222, row sums of A112798.

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.

Programs

Mathematica

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}]

Formula

a(n) = A257991 - A344616(n).

a(A122111(n)) = -a(n), where A122111 represents partition conjugation.