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Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).
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%I #33 May 23 2021 02:52:32

%S 0,0,1,0,2,1,4,2,7,6,13,11,22,22,38,39,63,69,103,114,165,189,262,301,

%T 407,475,626,733,950,1119,1427,1681,2118,2503,3116,3678,4539,5360,

%U 6559,7735,9400,11076,13372,15728,18886,22184,26501,31067,36947,43242,51210,59818,70576,82291,96750

%N Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

%C a(n) + A101708(n) = A064173(n).

%D George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

%H Alois P. Heinz, <a href="/A101707/b101707.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = (A000041(n) - A000025(n))/4. - _Vladeta Jovovic_, Dec 14 2004

%F G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - _Vladeta Jovovic_, Dec 20 2004

%F a(n) = A340692(n)/2. - _Gus Wiseman_, Feb 07 2021

%e a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).

%e From _Gus Wiseman_, Feb 07 2021: (Start)

%e Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:

%e 11 . 31 32 33 52 53 54 55

%e 1111 51 3211 71 72 73

%e 3111 3221 3222 91

%e 111111 3311 3321 3322

%e 5111 5211 3331

%e 311111 321111 5221

%e 11111111 5311

%e 7111

%e 322111

%e 331111

%e 511111

%e 31111111

%e 1111111111

%e Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:

%e 2 . 4 221 6 421 8 432 A

%e 211 222 22111 422 441 433

%e 411 431 621 442

%e 21111 611 22221 622

%e 22211 42111 631

%e 41111 2211111 811

%e 2111111 22222

%e 42211

%e 43111

%e 61111

%e 2221111

%e 4111111

%e 211111111

%e (End)

%p b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),

%p `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-

%p `if`(r<0, irem(i, 2), r))))

%p end:

%p a:= n-> b(n$2, -1)/2:

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Jan 29 2021

%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* _Gus Wiseman_, Feb 10 2021 *)

%t b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],

%t If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -

%t If[r < 0, Mod[i, 2], r]]]];

%t a[n_] := b[n, n, -1]/2;

%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 23 2021, after _Alois P. Heinz_ *)

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

%Y The even-rank version is A101708 (A340605).

%Y The even- but not necessarily positive-rank version is A340601 (A340602).

%Y The Heinz numbers of these partitions are (A340604).

%Y Allowing negative odd ranks gives A340692 (A340603).

%Y - Rank -

%Y A047993 counts balanced (rank zero) partitions (A106529).

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

%Y - Odd -

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

%Y A026804 counts partitions whose least part is odd.

%Y A027193 counts partitions of odd length/maximum (A026424/A244991).

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

%Y A339890 counts factorizations of odd length.

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

%Y Cf. A000041, A027187, A101709, A101199, A101200, A117409, A200750.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Dec 12 2004

%E More terms from _Joerg Arndt_, Oct 07 2012

%E a(0)=0 prepended by _Alois P. Heinz_, Jan 29 2021