A120452 Number of partitions of n-1 boys and one girl with no couple.

1, 1, 3, 5, 9, 14, 23, 34, 52, 75, 109, 153, 216, 296, 407, 549, 739, 981, 1300, 1702, 2224, 2879, 3716, 4761, 6083, 7721, 9774, 12306, 15450, 19307, 24064, 29867, 36978, 45614, 56130, 68846, 84250, 102793, 125148, 151955, 184123, 222553, 268482

Offset: 1

Author

Yasutoshi Kohmoto, Jul 20 2006

Keywords

• nonn
• easy

Comments

From Gus Wiseman, Jun 08 2021: (Start)

Also the number of:

- integer partitions of 2n with reverse-alternating sum 2;

- reversed integer partitions of 2n with alternating sum 2;

- integer partitions of 2n with exactly two odd parts, one of which is the greatest;

- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.

Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.

(End)

Examples

			n=5:
If partitions have no pair "o*", then a(5)=9 ("o" means a boy, "*" means a girl): {o, o, o, o, *}, {o, o, *, oo}, {*, oo, oo}, {o, *, ooo}, {o, o, oo*}, {oo, oo*}, {*, oooo}, {o, ooo*}, {oooo*}.
From _Gus Wiseman_, Jun 08 2021: (Start)
The a(1) = 1 through a(6) = 14 partitions of 2n with reverse-alternating sum 2:
  (2)  (211)  (222)    (332)      (442)        (552)
              (321)    (431)      (541)        (651)
              (21111)  (22211)    (22222)      (33222)
                       (32111)    (32221)      (33321)
                       (2111111)  (33211)      (43221)
                                  (43111)      (44211)
                                  (2221111)    (54111)
                                  (3211111)    (2222211)
                                  (211111111)  (3222111)
                                               (3321111)
                                               (4311111)
                                               (222111111)
                                               (321111111)
                                               (21111111111)
For example, the partition (43221) has reverse-alternating sum 1 - 2 + 2 - 3 + 4 = 2, so is counted under a(6).
The a(1) = 1 through a(6) = 14 partitions of 2n with exactly two odd parts, one of which is the greatest:
  (11)  (31)  (33)   (53)    (55)     (75)
              (51)   (71)    (73)     (93)
              (321)  (332)   (91)     (111)
                     (521)   (532)    (543)
                     (3221)  (541)    (552)
                             (721)    (732)
                             (3322)   (741)
                             (5221)   (921)
                             (32221)  (5322)
                                      (5421)
                                      (7221)
                                      (33222)
                                      (52221)
                                      (322221)
(End)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A000070, A002865.

A diagonal of A103919.

A diagonal of A344612.

A000097 counts partitions of 2n with alternating sum 2.

A001700/A088218 appear to count compositions with reverse-alternating sum 2.

A058696 counts partitions of 2n, ranked by A300061.

A344610 counts partitions of 2n by sum and positive reverse-alternating sum.

A344611 counts partitions of 2n with reverse-alternating sum >= 0.

A344741 counts partitions of 2n with reverse-alternating sum -2.

Cf. A001250, A027187, A119899, A124754, A239830, A316524, A325535, A344618, A344651, A344739.

Programs

Mathematica

a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021 *)

Formula

a(n) = A000070(n-2) + A002865(n-1). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006

a(n) = A000070(n-1) - A000041(n-2) = A000070(n-3) + A000041(n-1). - Max Alekseyev, Aug 23 2006

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 - 37*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Oct 25 2016

Extensions

More terms from Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006

More terms from Max Alekseyev, Aug 23 2006