A340692 Number of integer partitions of n of odd rank.

0, 0, 2, 0, 4, 2, 8, 4, 14, 12, 26, 22, 44, 44, 76, 78, 126, 138, 206, 228, 330, 378, 524, 602, 814, 950, 1252, 1466, 1900, 2238, 2854, 3362, 4236, 5006, 6232, 7356, 9078, 10720, 13118, 15470, 18800, 22152, 26744, 31456, 37772, 44368, 53002, 62134, 73894

Offset: 0

Author

Gus Wiseman, Jan 29 2021

Keywords

• nonn

Comments

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):
  .  .  (2)   .  (4)     (32)   (6)       (52)     (8)         (54)
        (11)     (31)    (221)  (33)      (421)    (53)        (72)
                 (211)          (51)      (3211)   (71)        (432)
                 (1111)         (222)     (22111)  (422)       (441)
                                (411)              (431)       (621)
                                (3111)             (611)       (3222)
                                (21111)            (3221)      (3321)
                                (111111)           (3311)      (5211)
                                                   (5111)      (22221)
                                                   (22211)     (42111)
                                                   (41111)     (321111)
                                                   (311111)    (2211111)
                                                   (2111111)
                                                   (11111111)
		

Links

• Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
• FindStat, St000145: The Dyson rank of a partition

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

The case of length/maximum instead of rank is A027193 (A026424/A244991).

The case of odd positive rank is A101707 is (A340604).

The strict case is A117193.

The even version is A340601 (A340602).

The Heinz numbers of these partitions are (A340603).

A072233 counts partitions by sum and length.

A168659 counts partitions whose length is divisible by maximum.

A200750 counts partitions whose length and maximum are relatively prime.

- Rank -

A047993 counts partitions of rank 0 (A106529).

A063995/A105806 count partitions by Dyson rank.

A064173 counts partitions of positive/negative rank (A340787/A340788).

A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).

A101198 counts partitions of rank 1 (A325233).

A101708 counts partitions of even positive rank (A340605).

A257541 gives the rank of the partition with Heinz number n.

A324520 counts partitions with rank equal to least part (A324519).

- Odd -

A000009 counts partitions into odd parts (A066208).

A026804 counts partitions whose least part is odd.

A058695 counts partitions of odd numbers (A300063).

A067659 counts strict partitions of odd length (A030059).

A160786 counts odd-length partitions of odd numbers (A300272).

A339890 counts factorizations of odd length.

A340385 counts partitions of odd length and maximum (A340386).

Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}]

Formula

Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0.