Peter J. Dukes - cp's OEIS frontend

User: Peter J. Dukes

Peter J. Dukes has authored 2 sequences.

A349216 Number of ternary triples (u,v,w) with 1 <= u < v < w <= n.

0, 0, 1, 2, 4, 8, 13, 20, 30, 40, 53, 70, 88, 110, 137, 166, 200, 240, 281, 328, 382, 438, 501, 572, 646, 728, 819, 910, 1010, 1120, 1233, 1356, 1490, 1628, 1777, 1938, 2100, 2274, 2461, 2652, 2856, 3074, 3297, 3534, 3786, 4040, 4309, 4594, 4884, 5190, 5513, 5842, 6188, 6552, 6917

Offset: 1

Peter J. Dukes, Nov 10 2021

A triple of integers (u,v,w) is a ternary triple if in the ternary expansions of u,v,w, all three disagree at the least significant position at which any two disagree.

Equivalently, (u,v,w) is a ternary triple if the highest power of three dividing 2w-u-v is greater than the highest power of three dividing gcd(w-u,w-v).

			For n = 7 the 13 ternary triples are (1, 2, 3), (2, 3, 4), (1, 3, 5), (3, 4, 5), (1, 2, 6), (2, 4, 6), (1, 5, 6), (4, 5, 6), (2, 3, 7), (1, 4, 7), (3, 5, 7), (2, 6, 7), (5, 6, 7).

Coen del Valle and Peter J. Dukes, Balancing permuted copies of multigraphs and integer matrices, arXiv:2201.00897 [math.CO], 2022.

Cf. A061866, A005704, A038500, A007089, A007997.

Array[Sum[Sum[Sum[Boole[IntegerExponent[w + w - u - v, 3] > IntegerExponent[GCD[w - u, w - v], 3]], {u, (v - 1)}], {v, 2, (w - 1)}], {w, 3, #}] &, 55] (* Michael De Vlieger, Feb 15 2022 *)

A349216(n) = sum(w=3,n,sum(v=2,(w-1),sum(u=1,(v-1),valuation(w+w-u-v,3) > valuation(gcd(w-u,w-v),3)))); \\ Antti Karttunen, Nov 13 2021

def a(n):
    t=3^ceil(log(n,3))
    counter=0
    for w in range(n):
        for v in range(w):
            for u in range(v):
                if min(gcd(w-u,3^t),gcd(w-v,3^t))

A246427 Number of facets of the cone defined by the zero-one inclusion matrix of pairs versus triples on an n-set.

Original entry on oeis.org

10, 70, 896, 52367

Offset: 5

Peter J. Dukes, Aug 26 2014

Equivalently, this is the number of integer weightings of the edges of the complete graph K_n which are: (1) nonnegative on all triangles; (2) maximally vanishing on triangles; and (3) have gcd of weights equal to one.

This also gives the degree of each anticut in the metric polytope (see link below) for n points.

			For n = 5, the 10 facet normals are defined by the choice of a (2,3)-partition.  Weight 2 is assigned to edges within each part and weight -1 is assigned to edges crossing the partition.  Every triangle has weight 0, except for one which inherits weight 6.

A. Deza, Metric Polytopes and Metric Cones
P. Dukes, Nearly complete count of isomorphism types for n = 9
P. Dukes and R. M. Wilson, The cone condition and t-designs, European J. Combin. 28 (2007), 1610-1625.
Peter J. Dukes, K. Garaschuk, On the cone of weighted graphs generated by triangles, arXiv preprint arXiv:1608.06017 [math.CO], 2016.
K. Garaschuk, Linear methods for rational triangle decompositions, Ph.D. Dissertation, University of Victoria, 2014.

Cf. A053043, A235459.

def A246427(n):
    T = Combinations(range(n),2)
    K = Combinations(range(n),3)
    W = matrix(ZZ,binomial(n,2),binomial(n,3),lambda i,j:Set(T[i]).issubset(Set(K[j])))
    C = Cone(W.transpose())
    return len(C.facet_normals())
[A246427(n) for n in range(5,8)]

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User: Peter J. Dukes

A349216 Number of ternary triples (u,v,w) with 1 <= u < v < w <= n.

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