A351981 Number 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

Offset

0,10

Links

Table of n, a(n) for n=0..55.

Example

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).

Mathematica

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}]

Crossrefs

The first condition alone is A277579, ranked by A349157.

The second condition alone is A277579, ranked by A350943.

These partitions are ranked by A351980.

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, A088218, A122111, A130780, A171966, A195017, A236559, A236914, A350849, A350942.

Sequence in context: A137992 A047654 A358477 * A058487 A062243 A128095

Adjacent sequences: A351978 A351979 A351980 * A351982 A351983 A351984

Keyword

nonn

Author

Gus Wiseman, Mar 15 2022

Status

approved