A350948 Number of integer partitions of n with as many even parts as even conjugate parts.

1, 1, 0, 3, 1, 5, 3, 7, 6, 10, 10, 18, 19, 27, 31, 40, 47, 65, 75, 98, 115, 142, 170, 217, 257, 316, 376, 458, 544, 671, 792, 952, 1129, 1351, 1598, 1919, 2259, 2681, 3155, 3739, 4384, 5181, 6064, 7129, 8331, 9764, 11380, 13308, 15477, 18047, 20944

Offset

0,4

Links

Table of n, a(n) for n=0..50.

Example

The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):
  ()  (1)  .  (3)    (22)  (5)      (42)    (7)        (62)
              (21)         (41)     (321)   (61)       (332)
              (111)        (311)    (2211)  (511)      (521)
                           (2111)           (4111)     (4211)
                           (11111)          (31111)    (32111)
                                            (211111)   (221111)
                                            (1111111)
For example, both (3,2,1,1,1) and its conjugate (5,2,1) have exactly 1 even part, so are counted under a(8).

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[#], _?EvenQ]&]], {n, 0, 30}]

Crossrefs

Comparing even to odd parts gives A045931, ranked by A325698.

The odd version is A277103, even rank case A345196, ranked by A350944.

Comparing even to odd conjugate parts gives A277579, ranked by A349157.

Comparing product of parts to product of conjugate parts gives A325039.

These partitions are ranked by A350945, the zeros of A350950.

A000041 counts integer partitions, strict A000009.

A103919 counts partitions by sum and alternating sum, reverse A344612.

A116482 counts partitions by number of even (or even conjugate) parts.

A122111 represents partition conjugation using Heinz numbers.

A257991 counts odd parts, conjugate A344616.

A257992 counts even parts, conjugate A350847.

A351976: # even = # even conj, # odd = # odd conj, ranked by A350949.

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

A351978: # even = # odd = # even conj = # odd conj, ranked by A350947.

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

Cf. A027187, A130780, A171966, A195017, A239241, A241638, A344607, A344651, A350848, A350941, A350942, A350943.

Sequence in context: A233526 A344674 A097062 * A324894 A200498 A227361

Adjacent sequences: A350945 A350946 A350947 * A350949 A350950 A350951

Keyword

nonn

Author

Gus Wiseman, Mar 14 2022

Status

approved