A340601 Number of integer partitions of n of even rank.

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Author

Gus Wiseman, Jan 21 2021

Keywords

• nonn

Comments

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

Examples

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• 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

Crossrefs

Note: Heinz numbers are given in parentheses below.

The positive case is A101708 (A340605).

The Heinz numbers of these partitions are A340602.

The odd version is A340692 (A340603).

- Rank -

A047993 counts partitions of rank 0 (A106529).

A072233 counts partitions by sum and length.

A101198 counts partitions of rank 1 (A325233).

A101707 counts partitions of odd positive rank (A340604).

A101708 counts partitions of even positive rank (A340605).

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

A340653 counts factorizations of rank 0.

- Even -

A024430 counts set partitions of even length.

A027187 counts partitions of even length (A028260).

A027187 (also) counts partitions of even maximum (A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A052841 counts ordered set partitions of even length.

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A236913 counts even-length partitions of even numbers (A340784).

A339846 counts factorizations of even length.

Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

Programs

Maple

b:= proc(n, i, r) option remember; `if`(n=0, 1-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):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Mathematica

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - 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];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

PARI

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

Formula

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024