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Number of integer partitions of n of odd rank.
17

%I #12 Apr 09 2021 09:41:08

%S 0,0,2,0,4,2,8,4,14,12,26,22,44,44,76,78,126,138,206,228,330,378,524,

%T 602,814,950,1252,1466,1900,2238,2854,3362,4236,5006,6232,7356,9078,

%U 10720,13118,15470,18800,22152,26744,31456,37772,44368,53002,62134,73894

%N Number of integer partitions of n of odd rank.

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

%H Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>

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

%e The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):

%e . . (2) . (4) (32) (6) (52) (8) (54)

%e (11) (31) (221) (33) (421) (53) (72)

%e (211) (51) (3211) (71) (432)

%e (1111) (222) (22111) (422) (441)

%e (411) (431) (621)

%e (3111) (611) (3222)

%e (21111) (3221) (3321)

%e (111111) (3311) (5211)

%e (5111) (22221)

%e (22211) (42111)

%e (41111) (321111)

%e (311111) (2211111)

%e (2111111)

%e (11111111)

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

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

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

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

%Y The strict case is A117193.

%Y The even version is A340601 (A340602).

%Y The Heinz numbers of these partitions are (A340603).

%Y A072233 counts partitions by sum and length.

%Y A168659 counts partitions whose length is divisible by maximum.

%Y A200750 counts partitions whose length and maximum are relatively prime.

%Y - Rank -

%Y A047993 counts partitions of rank 0 (A106529).

%Y A063995/A105806 count partitions by Dyson rank.

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

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

%Y A101198 counts partitions of rank 1 (A325233).

%Y A101708 counts partitions of even positive rank (A340605).

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

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

%Y - Odd -

%Y A000009 counts partitions into odd parts (A066208).

%Y A026804 counts partitions whose least part is odd.

%Y A058695 counts partitions of odd numbers (A300063).

%Y A067659 counts strict partitions of odd length (A030059).

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

%Y A339890 counts factorizations of odd length.

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 29 2021