arXiv subjects

Ian M. Wanless

Publications and source records attributed to Ian M. Wanless.

At least 37 records · Page 2Linked to original sources

arXiv2022

Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states

We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are arranged in a spatially symmetric pattern and the states have maximal entanglement for all bipartitions that follow from the reflection symmetries of the given geometry. We consider those cases where the state itself is invariant with respect to the geometrical symmetry group. The simplest examples are those dual unitary operators which are also self dual and reflection invariant, but we also consider the generalizations in the hexagonal, cubic, and octahedral geometries. We provide a number of constructions and concrete examples for these objects for various local dimensions. All of our examples can be used to build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple equivalent choices for the ``direction of time''.

quant-ph↗

arXiv2022

Pairs of MOLS of order ten satisfying non-trivial relations

A relation on a $k$-net$(n)$ (or, equivalently, a set of $k-2$ mutually orthogonal Latin squares of order $n$) is an $\mathbb{F}_{2}$ linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any 6-net(10) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are $18\,526\,320$ pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of $3$-MOLS from Dukes and Howard's classification.

math.CO↗

arXiv2022

On Perfect Sequence Covering Arrays

A PSCA$(v, t, λ)$ is a multiset of permutations of the $v$-element alphabet $\{0, \dots, v-1\}$ such that every sequence of $t$ distinct elements of the alphabet appears in the specified order in exactly $λ$ of the permutations. For $v \geq t \geq 2$, we define $g(v, t)$ to be the smallest positive integer $λ$ such that a PSCA$(v, t, λ)$ exists. We show that $g(6, 3) = g(7, 3) = g(7, 4) = 2$ and $g(8, 3) = 3$. Using suitable permutation representations of groups we make improvements to the upper bounds on $g(v, t)$ for many values of $v \leq 32$ and $3\le t\le 6$. We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.

math.CO↗

arXiv2022

Lieb's permanental dominance conjecture

We survey the impact of Lieb's influential paper "Proofs of some conjectures on permanents" [J. Math. Mech. 16 1966, 127-134], which introduced the famous permanental dominance conjecture. This conjecture has defied all attacks for over half a century, although a number of related conjectures have recently been resolved.

math.RT↗

arXiv2021

Small Latin arrays have a near transversal

A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an $n\times n$ array is a selection of $n$ cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order $n\le11$ has a diagonal of weight at least $n-1$. A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all $k\le20$ we compute a lower bound on the order of any Latin array that does not have a diagonal of weight at least $n-k$.

math.CO↗

arXiv2021

Enumeration of Latin squares with conjugate symmetry

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) The number of isomorphism classes of semisymmetric idempotent Latin squares of order $n$ equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order $n+1$, and (2) Suppose $A$ and $B$ are totally symmetric Latin squares of order $n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.

math.CO↗

arXiv2021

A general framework for hypergraph colouring

The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for colouring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colourings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colourings by Rosenfeld [2020]. We apply our general theorem in the setting of proper hypergraph colouring, proper graph colouring, independent transversals, star colouring, nonrepetitive colouring, frugal colouring, Ramsey number lower bounds, and for $k$-SAT.

math.CO↗

arXiv2021

Degree of Orthomorphism Polynomials over Finite Fields

An orthomorphism over a finite field $\mathbb{F}_q$ is a permutation $θ:\mathbb{F}_q\mapsto\mathbb{F}_q$ such that the map $x\mapstoθ(x)-x$ is also a permutation of $\mathbb{F}_q$. The degree of an orthomorphism of $\mathbb{F}_q$, that is, the degree of the associated reduced permutation polynomial, is known to be at most $q-3$. We show that this upper bound is achieved for all prime powers $q\notin\{2, 3, 5, 8\}$. We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct $3$-homogeneous Latin bitrades.

math.CO↗

arXiv2021

Existence results for cyclotomic orthomorphisms

An {\em orthomorphism} over a finite field $\mathbb{F}$ is a permutation $θ:\mathbb{F}\mapsto\mathbb{F}$ such that the map $x\mapstoθ(x)-x$ is also a permutation of $\mathbb{F}$. The orthomorphism $θ$ is {\em cyclotomic of index $k$} if $θ(0)=0$ and $θ(x)/x$ is constant on the cosets of a subgroup of index $k$ in the multiplicative group $\mathbb{F}^*$. We say that $θ$ has {\em least index} $k$ if it is cyclotomic of index $k$ and not of any smaller index. We answer an open problem due to Evans by establishing for which pairs $(q,k)$ there exists an orthomorphism over $\mathbb{F}_q$ that is cyclotomic of least index $k$. Two orthomorphisms over $\mathbb{F}_q$ are orthogonal if their difference is a permutation of $\mathbb{F}_q$. For any list $[b_1,\dots,b_n]$ of indices we show that if $q$ is large enough then $\mathbb{F}_q$ has pairwise orthogonal orthomorphisms of least indices $b_1,\dots,b_n$. This provides a partial answer to another open problem due to Evans. For some pairs of small indices we establish exactly which fields have orthogonal orthomorphisms of those indices. We also find the number of linear orthomorphisms that are orthogonal to certain cyclotomic orthomorphisms of higher index.

math.CO↗

arXiv2021

Zero-sum flows for Steiner systems

Given a $t$-$(v, k, λ)$ design, $\mathcal{D}=(X,\mathcal{B})$, a zero-sum $n$-flow of $\mathcal{D}$ is a map $f : \mathcal{B}\longrightarrow \{\pm1,\ldots, \pm(n-1)\}$ such that for any point $x\in X$, the sum of $f$ over all blocks incident with $x$ is zero. For a positive integer $k$, we find a zero-sum $k$-flow for an STS$(u w)$ and for an STS$(2v+7)$ for $v\equiv 1~(\mathrm{mod}~4)$, if there are STS$(u)$, STS$(w)$ and STS$(v)$ such that the STS$(u)$ and STS$(v)$ both have a zero-sum $k$-flow. In 2015, it was conjectured that for $v>7$ every STS$(v)$ admits a zero-sum $3$-flow. Here, it is shown that many cyclic STS$(v)$ have a zero-sum $3$-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.

math.CO↗

arXiv2020

Maximally nonassociative quasigroups via quadratic orthomorphisms

A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ only if $x=y=z$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2p_1$ or $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1\le p_2<2p_1$.

math.CO↗

arXiv2020

Induced path factors of regular graphs

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $ρ(G)$ of $G$ is the minimum number of paths in an induced path factor of $G$. We show that if $G$ is a connected cubic graph on $n>6$ vertices, then $ρ(G)\le(n-1)/3$. Fix an integer $k\ge3$. For each $n$, define $\mathcal{M}_n$ to be the maximum value of $ρ(G)$ over all connected $k$-regular graphs $G$ on $n$ vertices. As $n\rightarrow\infty$ with $nk$ even, we show that $c_k=\lim(\mathcal{M}_n/n)$ exists. We prove that $5/18\le c_3\le1/3$ and $3/7\le c_4\le1/2$ and that $c_k=\frac12-O(k^{-1})$ for $k\rightarrow\infty$.

math.CO↗

arXiv2020

Maximal sets of mutually orthogonal frequency squares

A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;λ)$ if it contains $n/λ$ symbols, each of which occurs $λ$ times per row and $λ$ times per column. In the case when $λ=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;λ)$ is a set of $k$ frequency squares of type $(n;λ)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;λ)$ is a set of $k$-MOFS$(n;λ)$ that is not contained in any set of $(k+1)$-MOFS$(n;λ)$. For even $n$, let $μ(n)$ be the smallest $k$ such that there exists a set of $k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7] that $μ(n)=1$ if $n/2$ is odd and $μ(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $μ(n)>2$. Also, we show that whenever $n$ is divisible by a particular function of $k$, there does not exist a set of $k'$-maxMOFS$(n;n/2)$ for any $k'\le k$. In particular, this means that $\limsup μ(n)$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides $n$. If $0\le v-uh<u/2$ for $h\ge1$ then we show that there exists a set of $(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.

math.CO↗

arXiv2020

Most binary matrices have no small defining set

Consider a matrix $M$ chosen uniformly at random from a class of $m \times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\mathit{defining}$ $\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\mathit{size}$ of a defining set is the number of filled entries. A $\mathit{critical}$ $\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $ε>0$, we assume that $n\le m=o(n^{1+ε})$, and that $λ\le1/2$, where $λ$ is the proportion of entries of $M$ that equal $1$. We also assume that the row sums of $M$ do not vary by more than $\mathcal{O}(n^{1/2+ε})$, and that the column sums do not vary by more than $\mathcal{O}(m^{1/2+ε})$. Under these assumptions we show that $M$ almost surely has no defining set of size less than $λmn-\mathcal{O}(m^{7/4+ε})$. It follows that $M$ almost surely has no critical set of size more than $(1-λ)mn+\mathcal{O}(m^{7/4+ε})$. Our results generalise a theorem of Cavenagh and Ramadurai, who examined the case when $λ=1/2$ and $n=m=2^k$ for an integer $k$.

math.CO↗

arXiv2020

On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups

Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z$ $\Leftrightarrow$ $x=y=z$. Denote by $σ(q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $α\approx 0.02908$ and $β\approx 0.01259$ such that if $q\equiv 1 \bmod 4$, then $\lim σ(q)/q^2 = α$, and if $q \equiv 3 \bmod 4$, then $\lim σ(q)/q^2 = β$.

math.CO↗

arXiv2020

Multidimensional permanents of polystochastic matrices

A $d$-dimensional matrix is called \emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a \emph{$1$-permutation} matrix. A \emph{diagonal} of a $d$-dimensional matrix of order $n$ is a choice of $n$ elements, no two in the same hyperplane. The \emph{permanent} of a $d$-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal. For a given order $n$ and dimension $d$, the set of $1$-polystochastic matrices forms a convex polytope that includes the $1$-permutation matrices within its set of vertices. For even $n$ and odd $d$, we give a construction for a class of $1$-permutation matrices with zero permanent. Consequently, we show that the set of $1$-polystochastic matrices with zero permanent contains at least $n^{n^{3/2}(1/2-o(1))}$ $1$-permutation matrices and contains a polytope of dimension at least $cn^{3/2}$ for fixed $c,d$ and even $n\to\infty$. We also provide counterexamples to a conjecture by Taranenko about the location of local extrema of the permanent. For odd $d$, we give a construction of $1$-permutation matrices that decompose into a convex linear sum of positive diagonals. These combine with a theorem of Taranenko to provide counterexamples to a conjecture by Dow and Gibson generalising van der Waerden's conjecture to higher dimensions.

math.CO↗

arXiv2019

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$.

math.CO↗

arXiv2019

Mutually orthogonal binary frequency squares

A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only {\em binary} frequency squares of order $n$ with $n/2$ zeroes and $n/2$ ones in each row and column. Two such frequency squares are \emph{orthogonal} if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A $k$-MOFS$(n)$ must satisfy $k\le(n-1)^2$, and any MOFS achieving this bound are said to be \emph{complete}. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2/4-O(n\log n)}$ isomorphism classes of complete MOFS$(n)$. For $2<n\equiv2\pmod4$ we show that there exists a $17$-MOFS$(n)$ but no complete MOFS$(n)$. A $k$-maxMOFS$(n)$ is a $k$-MOFS$(n)$ that is not contained in any $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a $k$-maxMOFS$(6)$ if and only if $k\in\{1,17\}$ or $5\le k\le 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $n\equiv2\pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $n\equiv0\pmod4$. We also prove that there exists a $5$-maxMOFS$(n)$ for each order $n\equiv 2\pmod{4}$ where $n\geq 6$.

math.CO↗