login
Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.
26

%I #13 Oct 18 2023 04:44:21

%S 1,1,1,1,2,3,2,4,4,5,5,8,9,11,11,16,16,20,20,25,30,34,38,42,50,58,64,

%T 73,80,90,105,114,128,148,158,180,201,220,241,277,306,333,366,404,447,

%U 497,544,592,662,708,797,861,954,1020,1131,1226,1352,1456,1600

%N Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

%e The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):

%e 1 2 3 4 5 6 7 8 9 A B C D E

%e 31 32 51 43 53 54 64 65 75 76 86

%e 41 52 62 72 73 74 93 85 95

%e 61 71 81 82 83 A2 94 A4

%e 531 91 92 B1 A3 B3

%e A1 543 B2 C2

%e 641 732 C1 D1

%e 731 741 652 851

%e 831 751 932

%e 832 941

%e 931 A31

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,15}]

%o (Python)

%o from collections import Counter

%o from itertools import combinations_with_replacement

%o from sympy.utilities.iterables import partitions

%o def A364346(n): return sum(1 for p in partitions(n) if max(p.values(),default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # _Chai Wah Wu_, Sep 20 2023

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

%Y For sums of any length > 1 we have A364349, non-strict A237667.

%Y The complement is counted by A363226, non-strict A363225.

%Y The non-strict version is A364345, ranks A364347, complement A364348.

%Y A000041 counts partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A236912 counts sum-free partitions not re-using parts, complement A237113.

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

%Y Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jul 22 2023