A363226 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.

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, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756

Offset: 0

Author

Gus Wiseman, Jul 19 2023

Keywords

• nonn

Comments

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

Examples

			The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
  21  .  .  42   421  431  63   532   542   84    643   653   A5
            321       521  432  541   632   642   742   743   843
                           621  631   821   651   841   752   942
                                721   5321  921   A21   761   C21
                                4321        5421  5431  842   6432
                                            6321  6421  B21   6531
                                                  7321  5432  7431
                                                        6431  7521
                                                        6521  8421
                                                        7421  9321
                                                        8321  54321
		

Crossrefs

For subsets of {1..n} we have A093971 (sum-full sets), complement A007865.

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

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

A000041 counts partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

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

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A025065, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083.

Programs

Mathematica

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

Python

from itertools import combinations_with_replacement
from collections import Counter
from sympy.utilities.iterables import partitions
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

Extensions

a(31)-a(56) from Chai Wah Wu, Sep 20 2023