A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750

Offset: 0

Author

Emeric Deutsch, Dec 12 2004

Keywords

• nonn

Comments

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

Examples

			a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
From _Gus Wiseman_, Feb 07 2021: (Start)
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:
  11   .  31     32   33       52     53         54       55
          1111        51       3211   71         72       73
                      3111            3221       3222     91
                      111111          3311       3321     3322
                                      5111       5211     3331
                                      311111     321111   5221
                                      11111111            5311
                                                          7111
                                                          322111
                                                          331111
                                                          511111
                                                          31111111
                                                          1111111111
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:
  2   .  4     221   6       421     8         432       A
         211         222     22111   422       441       433
                     411             431       621       442
                     21111           611       22221     622
                                     22211     42111     631
                                     41111     2211111   811
                                     2111111             22222
                                                         42211
                                                         43111
                                                         61111
                                                         2221111
                                                         4111111
                                                         211111111
(End)
		

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• FindStat, St000145: The Dyson rank of a partition

Crossrefs

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

The even-rank version is A101708 (A340605).

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

The Heinz numbers of these partitions are (A340604).

Allowing negative odd ranks gives A340692 (A340603).

- Rank -

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

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).

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

- Odd -

A000009 counts partitions into odd parts (A066208).

A026804 counts partitions whose least part is odd.

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

A058695 counts partitions of odd numbers (A300063).

A339890 counts factorizations of odd length.

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

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

Programs

Maple

b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1)/2:
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 29 2021

Mathematica

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
     If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
     If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
a /@ Range[0, 55] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

Formula

a(n) = (A000041(n) - A000025(n))/4. - Vladeta Jovovic, Dec 14 2004

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

a(n) = A340692(n)/2. - Gus Wiseman, Feb 07 2021

Extensions

More terms from Joerg Arndt, Oct 07 2012

a(0)=0 prepended by Alois P. Heinz, Jan 29 2021