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Number of integer partitions of n with no adjacent parts of quotient 2.
11

%I #12 Jan 25 2022 10:26:10

%S 1,1,2,2,4,5,7,10,14,18,24,31,41,53,70,87,112,140,178,221,277,344,428,

%T 526,648,792,971,1180,1436,1738,2103,2533,3049,3660,4387,5242,6259,

%U 7450,8860,10511,12453,14723,17387,20489,24121,28343,33269,38982,45632,53327

%N Number of integer partitions of n with no adjacent parts of quotient 2.

%C The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.

%e The a(1) = 1 through a(7) = 10 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (111) (22) (32) (33) (43)

%e (31) (41) (51) (52)

%e (1111) (311) (222) (61)

%e (11111) (411) (322)

%e (3111) (331)

%e (111111) (511)

%e (4111)

%e (31111)

%e (1111111)

%t Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]

%Y The version with quotients >= 2 is A000929, sets A018819.

%Y <= 2 is A342094, ranked by A342191.

%Y < 2 is A342096, sets A045690, strict A342097.

%Y > 2 is A342098, sets A040039.

%Y The sets version (subsets of prescribed maximum) is A045691.

%Y These partitions are ranked by A350838.

%Y The strict case is A350840.

%Y A version for differences is A350842, strict A350844.

%Y The complement is counted by A350846, ranked by A350845.

%Y A000041 = integer partitions.

%Y A116931 = partitions with no successions, ranked by A319630.

%Y A116932 = partitions with differences != 1 or 2, strict A025157.

%Y A323092 = double-free partitions, ranked by A320340.

%Y Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 18 2022