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

0,3

Comments

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

Links

Table of n, a(n) for n=0..48.

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

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.

Example

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)

Mathematica

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

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.

Sequence in context: A278082 A327442 A068773 * A234312 A244136 A338212

Adjacent sequences: A340689 A340690 A340691 * A340693 A340694 A340695

Keyword

nonn

Author

Gus Wiseman, Jan 29 2021

Status

approved