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The balancing index of a polynomial $f \in \mathbb{Z}[x_1,\dots,x_n]$ is the least positive sum of coefficients in an integer linear combination of permuted copies of $f$ which produces a symmetric polynomial. Here we consider the restricted problem in which only even permutations are used.

A Steiner triple system, STS$(v)$, is a family of $3$-subsets (blocks) of a set of $v$ elements such that any two elements occur together in precisely one block. A collection of triples consisting of two copies of each block of an STS is called a duplicated Steiner triple system, DSTS. A resolvable (or near resolvable) DSTS is called self-orthogonal if every pair of distinct classes in the resolution has at most one block in common. We provide several methods to construct self-orthogonal near resolvable DSTS and settle the existence of such designs for all values of $v$ with only four possible exceptions. This addresses a recent question of Bryant, Davies and Neubecker.

We study a system of linear equations associated with Sudoku latin squares. The coefficient matrix $M$ of the normal system has various symmetries arising from Sudoku. From this, we find the eigenvalues and eigenvectors of $M$, and compute a generalized inverse. Then, using linear perturbation methods, we obtain a fractional completion guarantee for sufficiently large and sparse rectangular-box Sudoku puzzles.

A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and only if the density of each permutation $\sigma$ in the set $$\{123,321,2143,3412,2413,3142\}$$ converges to $1/|\sigma|!$. Previously, the smallest cardinality of a set with this property, called a "quasirandom-forcing" set, was known to be between four and eight. In fact, we show that there is a single linear expression of the densities of the six permutations in this set which forces quasirandomness and show that this is best possible in the sense that there is no shorter linear expression of permutation densities with positive coefficients with this property. In the language of theoretical statistics, this expression provides a new nonparametric independence test for bivariate continuous distributions related to Spearman's $\rho$., Comment: 27 pages, 1 figure, 5 appendices included as ancillary files

For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every vertex of $G$ is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the $3$-neighbour bootstrap process when $G$ is a $3$-dimensional grid with minimum side-length at least $11$. We also characterize the integers $a$ and $b$ for which there is a set of cardinality $\frac{ab+a+b}{3}$ that percolates for the $3$-neighbour bootstrap process in the $a\times b$ grid; this solves a problem raised by Benevides, Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021]., Comment: 45 pages

Given a family $\mathcal{F}=\{A_1,\dots,A_s\}$ of subsets of $\mathbb{Z}_n$, define $\Delta \mathcal{F}$ to be the multiset of all (cyclic) distances dist$(x,y)$, where $\{x,y\} \subset A_i$, $x \neq y$, for some $i=1,\dots,s$. Taking inspiration from a Euclidean distance problem of Erd\H{o}s, we say that $\mathcal{F}$ is Erd\H{o}s-deep if the multiplicities of distances that occur in $\Delta \mathcal{F}$ are precisely $1,2,\dots,k-1$ for some integer $k$. In the case $s=1$, it is known that a modular arithmetic progression in $\mathbb{Z}_n$ achieves this property (under mild conditions); conversely, APs are the only such sets, except for one sporadic case when $n=6$. Here, we consider in detail the case $s=2$. In particular, we classify Erd\H{o}s-deep pairs $\{A_1,A_2\}$ when each $A_i$ is an arithmetic progression in $\mathbb{Z}_n$. We also give a construction of a much wider class of Erd\H{o}s-deep families $\{A_1,\dots,A_s\}$ when $s$ is a square integer.

Given a square matrix $A$ over the integers, we consider the $\mathbb{Z}$-module $M_A$ generated by the set of all matrices that are permutation-similar to $A$. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices $a I + b J$ belonging to $M_A$. We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over $\mathbb{Z}$. We consider several special cases in detail. In particular, the problem for symmetric matrices answers a question of Cameron and Cioab\v{a} on determining the eventual period for integers $\lambda$ such that the $\lambda$-fold complete graph $\lambda K_n$ has an edge-decomposition into a given (multi)graph.

A balanced incomplete block design is a set system in which all pairs of distinct elements occur with a constant frequency. By contrast, a Sarvate-Beam design induces an interval of distinct frequencies on pairs. In this paper, we settle the existence of a Sarvate-Beam variant of group divisible designs of uniform type with block size three.

A Kirkman triple system of order $v$, KTS$(v)$, is a resolvable Steiner triple system on $v$ elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS$(v)$ which contain as a subdesign a Steiner triple system of order $u$, an STS$(u)$. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case $v=2u+1$, for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case $v=2u+3$ to now three possible exceptions. In addition, we obtain results for other cases of the form $v=2u+w$ and also near $v=3u$. Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.

We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix $B \in \mathbb{F}^{n \times n}$ is defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_j \in \{\tt{A}, \tt{S}, \tt{N}\}$ according to whether all, some but not all, or none of the principal minors of order $j$ of $B$ are nonzero. Building upon the second author's recent classification of the epr-sequences of symmetric matrices over the field $\mathbb{F}=\mathbb{F}_2$, we initiate a study of the case $\mathbb{F}=\mathbb{F}_3$. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.

It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which denotes the maximum number of `HMOLS' or mutually orthogonal latin squares having a common equipartition into $n$ holes of a fixed size $h$. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound $N(h^n) \ge (\log n)^{1/\delta}$ for any $\delta>2$ and all $n > n_0(h,\delta)$.

In a balanced graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local balance conditions in graph decompositions. In the basic case where a single graph with colored loops is used as a block, an existence theory for such decompositions follows as a straightforward generalization of previous work on balanced graph decompositions.

We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n, \text{diag}(S_k) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections., Comment: 18 pages

We prove that, for sufficiently large $n$, every graph of order $n$ with minimum degree at least $0.852n$ has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross to establish a bound of $0.9n$. By a result of Barber, K\"{u}hn, Lo and Osthus, our result implies that, for each $\epsilon >0$, every graph of sufficiently large order $n$ with minimum degree at least $(0.852+\epsilon)n$ has a triangle decomposition if and only if it has all even degrees and number of edges a multiple of three., Comment: 15 pages, 0 figures

The dimension of a block design is the maximum positive integer $d$ such that any $d$ of its points are contained in a proper subdesign. Pairwise balanced designs PBD$(v,K)$ have dimension at least two as long as not all points are on the same line. On the other hand, designs of dimension three appear to be very scarce. We study designs of dimension three with block sizes in $K=\{3,4\}$ or $\{3,5\}$, obtaining several explicit constructions and one nonexistence result in the latter case. As applications, we obtain a result on dimension three triple systems having arbitrary index as well as symmetric latin squares which are covered in a similar sense by proper subsquares.

We consider maximum packings of edge-disjoint $4$-cliques in the complete graph $K_n$. When $n \equiv 1$ or $4 \pmod{12}$, these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible `leave' graphs induced by those edges. We give particular emphasis to the case $n \equiv 0$ or $3 \pmod{12}$, when the leave is $2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of $2$-regular leaves. For various subsets $S \subseteq \{3,4,5,\dots\}$, we establish explicit lower bounds on $n$ to guarantee the existence of maximum packings with any possible leave whose cycle lengths belong to $S$., Comment: 19 pages plus supplementary file

A classical recursive construction for mutually orthogonal latin squares (MOLS) is shown to hold more generally for a class of permutation codes of length $n$ and minimum distance $n-1$. When such codes of length $p+1$ are included as ingredients, we obtain a general lower bound $M(n,n-1) \ge n^{1.079}$ for large $n$, gaining a small improvement on the guarantee given from MOLS.

We give explicit constructions for incomplete pairwise balanced designs IPBD$((v;w),K)$, or, equivalently, edge-decompositions of a difference of two cliques $K_v \setminus K_w$ into cliques whose sizes belong to the set $K$. Our constructions produce such designs whenever $v$ and $w$ satisfy the usual divisibility conditions, have ratio $v/w$ bounded away from the smallest value in $K$ minus one, say $v/w > k-1+\epsilon$, for $k =\min K$ and $\epsilon>0$, and are sufficiently large (depending on $K$ and $\epsilon$). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as `templates'.

An $(n_3)$ configuration is an incidence structure equivalent to a linear hypergraph on $n$ vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz' theorem for this setting.

Motivated by problems involving triangle-decompositions of graphs, we examine the facet structure of the cone $\tau_n$ of weighted graphs on $n$ vertices generated by triangles. Our results include enumeration of facets for small $n$, a construction producing facets of $\tau_{n+1}$ from facets of $\tau_n$, and an arithmetic condition on entries of the normal vectors. We also point out that a copy of $\tau_n$ essentially appears via the perimeter inequalities at one vertex of the metric polytope.

We compute a minimum degree threshold sufficient for 3-partite graphs to admit a fractional triangle decomposition. Together with recent work of Barber, K\"uhn, Lo, Osthus and Taylor, this leads to bounds for exact decompositions and in particular the completion problem for sparse partial latin squares. Some extensions are considered as well.

We prove that for any $K$ and $d$, there exist, for all sufficiently large admissible $v$, a pairwise balanced design PBD$(v,K)$ of dimension $d$ for which all $d$-point-generated flats are bounded by a constant independent of $v$. We also tighten a prior upper bound for $K = \{3,4,5\}$, in which case there are no divisibility restrictions on the number of points. One consequence of this latter result is the construction of latin squares `covered' by small subsquares.

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$.

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular $N_2$ (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near $\binom{v}{2}$ is obtained for large orders $v$.

Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system $\mathcal{F} \subseteq 2^X$ is $t$-laminar if $A,B \in \mathcal{F}$ with $|A \cap B| \ge t$ implies $A \subseteq B$ or $B \subseteq A$. We obtain very close asymptotic bounds in terms of $n$ on the maximum size of a $2$-laminar family $\mathcal{F} \subseteq 2^{[n]}$. A construction for $3$-laminar families and a crude analysis for general $t$ are also given.

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, `triple products' of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.

The dimension of a linear space is the maximum positive integer $d$ such that any $d$ of its points generate a proper subspace. For a set $K$ of integers at least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space on $v$ points whose lines (or blocks) have sizes belonging to $K$. We show that, for any prescribed set of sizes $K$ and lower bound $d$ on the dimension, there exists a PBD$(v,K)$ of dimension at least $d$ for all sufficiently large and numerically admissible $v$.

We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym$(n)$. In particular, we compute orbits of ordered pairs on Sym$(n)$ acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size $M(n,d)$ of permutation codes of lengths up to 7. For instance, these techniques detect the nonexistence of the projective plane of order six via $M(6,5)<30$ and yield a new best bound $M(7,4) \le 535$ for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results., Comment: 13 pages, 4 tables

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur with different frequencies, such a family has been called (by Sarvate and others) a $t$-adesign. An elementary observation shows that such families always exist for $v > k \ge t$. Here, we study the smallest possible maximum frequency $\mu=\mu(t,k,v)$. The exact value of $\mu$ is noted for $t=1$ and an upper bound (best possible up to a constant multiple) is obtained for $t=2$ using PBD closure. Weaker, yet still reasonable asymptotic bounds on $\mu$ for higher $t$ follow from a probabilistic argument. Some connections are made with the famous Sidon problem of additive number theory., Comment: 6 pages

Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, although no concrete understanding of `finitely many' is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$. Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$., Comment: 17 pages

We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently many vertices and `local density' at least $1-C(t)k^{-2t}$. In the case $t=2$ and $H=K_3$, we show that all large graphs with density at least $27/28$ admit a fractional triangle decomposition. The proof relies on estimates of certain eigenvalues in the Johnson scheme.