A351976 Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 9, 11, 11, 16, 21, 22, 24, 31, 41, 46, 48, 64, 82, 91, 98, 120, 155, 175, 188, 237, 297, 329, 357, 437, 544, 607, 658, 803, 987, 1098, 1196, 1432, 1749, 1955, 2126, 2541, 3071, 3417, 3729, 4406, 5291, 5890, 6426

Offset

0,9

Links

Table of n, a(n) for n=0..58.

Example

The a(n) partitions for selected n:
n = 3     8       11        12        15          16
   ----------------------------------------------------------
    (21)  (332)   (4322)    (4332)    (4443)      (4444)
          (4211)  (4331)    (4422)    (54321)     (53332)
                  (4421)    (4431)    (632211)    (55222)
                  (611111)  (53211)   (633111)    (55411)
                            (621111)  (642111)    (633211)
                                      (81111111)  (642211)
                                                  (643111)
                                                  (7321111)
                                                  (82111111)

Mathematica

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]==Count[conj[#], _?OddQ]&&Count[#, _?EvenQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]

Crossrefs

The first condition alone is A277103, ranked by A350944, strict A000700.

The second condition alone is A350948, ranked by A350945.

These partitions are ranked by A350949.

A000041 counts integer partitions.

A122111 represents partition conjugation using Heinz numbers.

A195017 = # of even parts - # of odd parts.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four other possible pairings of statistics:

- A045931: # even = # odd, ranked by A325698, strict A239241.

- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.

- A277579: # even = # odd conj, ranked by A349157, strict A352131.

- A277579: # even conj = # odd, ranked by A350943, strict A352130.

There are two other possible double-pairings of statistics:

- A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.

- A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.

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

Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.

Sequence in context: A122385 A035002 A032578 * A035659 A008282 A296690

Adjacent sequences: A351973 A351974 A351975 * A351977 A351978 A351979

Keyword

nonn

Author

Gus Wiseman, Mar 14 2022

Status

approved