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Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.
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%I #18 Oct 18 2023 04:44:04

%S 0,0,0,1,0,0,2,1,2,3,5,4,6,7,11,11,16,18,26,29,34,42,51,62,72,84,101,

%T 119,142,166,191,226,262,300,354,405,467,540,623,705,807,927,1060,

%U 1206,1369,1551,1760,1998,2248,2556,2861,3236,3628,4100,4587,5152,5756

%N Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.

%C Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

%e The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):

%e 21 . . 42 421 431 63 532 542 84 643 653 A5

%e 321 521 432 541 632 642 742 743 843

%e 621 631 821 651 841 752 942

%e 721 5321 921 A21 761 C21

%e 4321 5421 5431 842 6432

%e 6321 6421 B21 6531

%e 7321 5432 7431

%e 6431 7521

%e 6521 8421

%e 7421 9321

%e 8321 54321

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

%o (Python)

%o from itertools import combinations_with_replacement

%o from collections import Counter

%o from sympy.utilities.iterables import partitions

%o def A363226(n): return sum(1 for p in partitions(n) if max(p.values(),default=0)==1 and 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 A093971 (sum-full sets), complement A007865.

%Y The non-strict version is A363225, ranks A364348 (complement A364347).

%Y The complement is counted by A364346, non-strict A364345.

%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, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083.

%K nonn

%O 0,7

%A _Gus Wiseman_, Jul 19 2023

%E a(31)-a(56) from _Chai Wah Wu_, Sep 20 2023