proposed

approved

editing

proposed

Cf. A000070, A088218, A122111, A130780, A171966, A195017, A236559, A236914, A350849, A350941, A350942.

The conjugate version is the same A351981.

Cf. `A000070, ~A026424, ~A028260, A088218, A122111, A130780, A171966, A195017, A236559, A236914, `A325700, `A350839, A350849, `A350941, A350942, `A350950, `A350951.

allocated for Gus WisemanNumber of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.

1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 2, 2, 4, 2, 1, 6, 8, 7, 9, 13, 14, 15, 19, 21, 23, 32, 40, 41, 45, 66, 81, 80, 96, 124, 139, 160, 194, 221, 246, 303, 360, 390, 446, 546, 634, 703, 810, 971, 1115, 1250, 1448, 1685, 1910

0,10

The a(n) partitions for selected n:

n = 3 9 15 18 19 20 21

-----------------------------------------------------------

21 4221 622221 633222 633322 644321 643332

4311 632211 643221 643321 653321 654321

642111 643311 644221 654221 665211

651111 644211 644311 654311 82222221

653211 653221 82222211 83222211

663111 653311 84221111 84222111

654211 86111111 85221111

664111 86211111

87111111

For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];

Table[Length[Select[IntegerPartitions[n], Count[#, _?EvenQ]==Count[conj[#], _?OddQ]&&Count[#, _?OddQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]

The first condition alone is A277579, ranked by A349157.

The second condition alone is A277579, ranked by A350943.

These partitions are ranked by A351980.

The conjugate version is the same A351981.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four other pairings of statistics:

- A045931: # of even parts = # of odd parts:

- conjugate also A045931

- ordered A098123

- strict A239241

- ranked by A325698

- conjugate ranked by A350848

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

- A350948: # of even parts = # of even conjugate parts, ranked by A350945.

There are two other double-pairings of statistics:

- A351976, ranked by A350949.

- A351977, ranked by A350946.

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

Cf. `A000070, ~A026424, ~A028260, A088218, A122111, A130780, A171966, A195017, A236559, A236914, `A325700, `A350839, A350849, `A350941, A350942, `A350950, `A350951.

allocated

nonn

Gus Wiseman, Mar 15 2022

approved

editing

allocated for Gus Wiseman

allocated

approved