Henry L. Fleischmann - cp's OEIS frontend

User: Henry L. Fleischmann

Henry L. Fleischmann has authored 2 sequences.

A349775 The maximum cardinality of an irreducible subset of {0, 1, 2, ..., n}.

2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24

Offset: 1

Henry L. Fleischmann, Nov 29 2021

A subset S of {0,1,2, ... n} is "irreducible" if it cannot be written as A + B = S for any A and B subsets of the integers with |A|, |B| >= 2. In this case, A + B is defined as the set of a + b for a in A and b in B.

Irreducible sets are also sometimes referred to as "Ostmann irreducible," "prime," or "primitive".

			{0,1,2} is reducible since {0,1} + {0,1} = {0,1,2}. All other subsets of {0,1,2} are irreducible, so a(2) = 2. It is possible to reduce to the case of assuming every set contains only nonnegative integers and each set contains 0 by shifting the summand sets.

M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Subsets, Springer 1996.
H. H. Ostmann, Additive Zahlentheorie, Springer 1956.

Henry L. Fleischmann, Python Code

Python
```
# See Fleischmann link.
```

from itertools import combinations
from collections import Counter
from math import comb
def A349775sumsetgen(n): # generate sums of 2 subsets A,B with |A|,|B| >= 2
    for l in range(2,n+2):
        for a in combinations(range(n+1),l):
            amax = max(a)
            bmax = min(amax,n-amax)
            for lb in range(2,bmax+2):
                for b in combinations(range(bmax+1),lb):
                    yield tuple(sorted(set(x+y for x in a for y in b)))
def A349775(n):
    c = Counter()
    for s in set(A349775sumsetgen(n)):
        c[len(s)] += 1
    for i in range(n+1,1,-1):
        if c[i] < comb(n+1,i):
            return i # Chai Wah Wu, Dec 14 2021

a(25)-a(27) from Chai Wah Wu, Dec 14 2021

a(28) from Chai Wah Wu, Dec 17 2021

A346796 Number of equivalence classes of triangles in an n-dimensional hypercube, equivalent up to translation of difference vectors corresponding to edges.

Original entry on oeis.org

0, 2, 22, 180, 1340, 9622, 68082, 478760, 3357880, 23524842, 164732942, 1153307740, 8073685620, 56517393662, 395626538602, 2769400119120, 19385843880560, 135701036304082, 949907641549062, 6649354653104900

Offset: 1

Henry L. Fleischmann, Aug 04 2021

Proved via a combinatorial argument.

			The 1-dimensional hypercube (vertices 0 and 1 on a line) has no triangles and thus no classes of triangle equivalent up to edge translation, so a(1)=0.
A square, the 2-dimensional hypercube, has two distinct right triangles up to edge translation, so a(2)=2.

Henry L. Fleischmann et al., Distinct Angle Problems and Variants, arXiv:2108.12015 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Cf. A016212 (allowing flips as well as edge translations, up to offset).

def a(n): return (7**n - 3**(n+1) + 2)//12

a(n) = (7^n - 3^(n+1) + 2)/12.
a(n) = 2*A016212(n-2) for n >= 2.
G.f.: 2*x^2/(1 - 11*x + 31*x^2 - 21*x^3). - Stefano Spezia, Aug 04 2021

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User: Henry L. Fleischmann

A349775 The maximum cardinality of an irreducible subset of {0, 1, 2, ..., n}.

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A346796 Number of equivalence classes of triangles in an n-dimensional hypercube, equivalent up to translation of difference vectors corresponding to edges.

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