login
Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.
37

%I #15 Oct 18 2023 04:43:41

%S 1,1,2,2,4,5,7,10,13,16,21,27,34,43,54,67,83,102,122,151,182,218,258,

%T 313,366,443,513,611,713,844,975,1149,1325,1554,1780,2079,2381,2761,

%U 3145,3647,4134,4767,5408,6200,7014,8035,9048,10320,11639,13207,14836,16850

%N Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

%e The a(1) = 1 through a(8) = 13 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (32) (33) (43) (44)

%e (31) (41) (51) (52) (53)

%e (1111) (311) (222) (61) (62)

%e (11111) (411) (322) (71)

%e (3111) (331) (332)

%e (111111) (511) (611)

%e (4111) (2222)

%e (31111) (3311)

%e (1111111) (5111)

%e (41111)

%e (311111)

%e (11111111)

%t Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]

%Y For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.

%Y Without re-using parts we have A236912, complement A237113.

%Y Allowing the sum of any number of parts gives A237667 (cf. A108917).

%Y The complement is counted by A363225, strict A363226, for subsets A093971.

%Y The strict case is A364346.

%Y These partitions have ranks A364347, complement A364348.

%Y A000041 counts partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A323092 counts double-free partitions, ranks A320340.

%Y Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 20 2023