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%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
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