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Number of partitions of n-1 boys and one girl with no couple.
45

%I #24 Jun 11 2021 18:25:49

%S 1,1,3,5,9,14,23,34,52,75,109,153,216,296,407,549,739,981,1300,1702,

%T 2224,2879,3716,4761,6083,7721,9774,12306,15450,19307,24064,29867,

%U 36978,45614,56130,68846,84250,102793,125148,151955,184123,222553,268482

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

%C From _Gus Wiseman_, Jun 08 2021: (Start)

%C Also the number of:

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

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

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

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

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

%C (End)

%H Vaclav Kotesovec, <a href="/A120452/b120452.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = A000070(n-1) - A000041(n-2) = A000070(n-3) + A000041(n-1). - _Max Alekseyev_, Aug 23 2006

%F 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

%e n=5:

%e 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*}.

%e From _Gus Wiseman_, Jun 08 2021: (Start)

%e The a(1) = 1 through a(6) = 14 partitions of 2n with reverse-alternating sum 2:

%e (2) (211) (222) (332) (442) (552)

%e (321) (431) (541) (651)

%e (21111) (22211) (22222) (33222)

%e (32111) (32221) (33321)

%e (2111111) (33211) (43221)

%e (43111) (44211)

%e (2221111) (54111)

%e (3211111) (2222211)

%e (211111111) (3222111)

%e (3321111)

%e (4311111)

%e (222111111)

%e (321111111)

%e (21111111111)

%e For example, the partition (43221) has reverse-alternating sum 1 - 2 + 2 - 3 + 4 = 2, so is counted under a(6).

%e The a(1) = 1 through a(6) = 14 partitions of 2n with exactly two odd parts, one of which is the greatest:

%e (11) (31) (33) (53) (55) (75)

%e (51) (71) (73) (93)

%e (321) (332) (91) (111)

%e (521) (532) (543)

%e (3221) (541) (552)

%e (721) (732)

%e (3322) (741)

%e (5221) (921)

%e (32221) (5322)

%e (5421)

%e (7221)

%e (33222)

%e (52221)

%e (322221)

%e (End)

%t a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];

%t Array[a, 50] (* _Jean-François Alcover_, Jun 05 2021 *)

%Y Cf. A000041, A000070, A002865.

%Y A diagonal of A103919.

%Y A diagonal of A344612.

%Y A000097 counts partitions of 2n with alternating sum 2.

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

%Y A058696 counts partitions of 2n, ranked by A300061.

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

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

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

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

%K nonn,easy

%O 1,3

%A _Yasutoshi Kohmoto_, Jul 20 2006

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

%E More terms from _Max Alekseyev_, Aug 23 2006