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Number of strict integer partitions of n whose length is not 2.
5

%I #6 Sep 21 2023 08:56:36

%S 1,1,1,1,1,1,2,2,3,4,6,7,10,12,16,20,25,30,38,45,55,66,79,93,111,130,

%T 153,179,209,242,282,325,375,432,496,568,651,742,846,963,1094,1240,

%U 1406,1589,1795,2026,2282,2567,2887,3240,3634,4072,4557,5094,5692,6351

%N Number of strict integer partitions of n whose length is not 2.

%C Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

%F a(n) = A000009(n) - A004526(n-1) for n > 0.

%e The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):

%e (5) (6) (7) (8) (9) (A) (B) (C) (D)

%e (321) (421) (431) (432) (532) (542) (543) (643)

%e (521) (531) (541) (632) (642) (652)

%e (621) (631) (641) (651) (742)

%e (721) (731) (732) (751)

%e (4321) (821) (741) (832)

%e (5321) (831) (841)

%e (921) (931)

%e (5421) (A21)

%e (6321) (5431)

%e (6421)

%e (7321)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

%Y The complement is counted by A140106 shifted left.

%Y Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.

%Y The non-strict version is A058984, complement A004526.

%Y The case not containing n/2 is A365826, non-strict A365825.

%Y A000041 counts integer partitions, strict A000009.

%Y A046663 counts partitions with no submultiset summing to k, strict A365663.

%Y A182616 counts partitions of 2n that do not contain n, strict A365828.

%Y Cf. A008967, A035363, A078408, A365659.

%K nonn

%O 0,7

%A _Gus Wiseman_, Sep 20 2023