A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

0, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, 0, -3, 0, 3, 1, 1, 0, 0, 0, -1, -1, -2, 0, 1, 0, 0, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, -2, -1, 1, 1, 2, -3, 0, 0, 0, 0, -1, 1, -1, 3, -1, -2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -3, 1, 1, 0

Offset

1,25

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.

Links

Table of n, a(n) for n=1..80.

Formula

a(n) = A257992(n) - A350847(n).

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

Example

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

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], _?EvenQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}]

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 A350945, counted by A350948.

The version comparing odd with odd conjugate parts is A350951.

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.

- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

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.

A116482 counts partitions by number of even parts.

A122111 represents partition conjugation using Heinz numbers.

A316524 gives the alternating sum of prime indices.

Cf. A028260, A088218, A098123, A236559, A239241, A241638, A325700, A347450.

Sequence in context: A328382 A211868 A127372 * A051171 A336118 A342639

Adjacent sequences: A350947 A350948 A350949 * A350951 A350952 A350953

Keyword

sign

Author

Gus Wiseman, Mar 14 2022

Status

approved