16 research outputs found
Two classes of minimal generic fundamental invariants for tensors
Motivated by the problems raised by B\"{u}rgisser and Ikenmeyer, we discuss
two classes of minimal generic fundamental invariants for tensors of order 3.
The first one is defined on , where . We study
its construction by obstruction design introduced by B\"{u}rgisser and
Ikenmeyer, which partially answers one problem raised by them. The second one
is defined on . We study its evaluation on the matrix multiplication
tensor and unit tensor when
. The evaluation on the unit tensor leads to the definition of Latin
cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin
square to Latin cube, which enrich the understanding of 3-dimensional
Alon-Tarsi problem. It is also natural to generalize the constructions to
tensors of other orders. We illustrate the distinction between even and odd
dimensional generalizations by concrete examples. Finally, some open problems
in related fields are raised.Comment: Some typos were changed.New publication information has been update
There are asymptotically the same number of Latin squares of each parity
A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order n there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order n → ∞
Coloring linear hypergraphs: the Erdos-Faber-Lovasz conjecture and the Combinatorial Nullstellensatz
The long-standing Erdos-Faber-Lovasz conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdos-Faber-Lovasz conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work
On the problem of Molluzzo for the modulus 4
We solve the currently smallest open case in the 1976 problem of Molluzzo on
, namely the case . This amounts to constructing,
for all positive integer congruent to or , a sequence of
integers modulo of length generating, by Pascal's rule, a Steinhaus
triangle containing with equal multiplicities.Comment: 12 pages ; 3 figures ; 3 tables, Integers : Electronic Journal of
Combinatorial Number Theory, State University of West Georgia, Charles
University, and DIMATIA, 2012, 12, pp.A1
Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups
An partial Latin rectangle is an matrix containing elements of such that each row and each column contain at most one copy of any symbol in . An entry is a triple with . Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of -entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) , i.e., partial Latin squares, (b) and , and (c) and