We survey here results and problems from the reconstruction theory of evolutionary trees, which involve enumeration and inversion.

It is proved in this paper that for any integer $n\ge 100$, a $(v,n)$-IODLS (incomplete orthogonal diagonal Latin squares) exists if and only if $v\ge 3n+2$. Results for $n=2$ are also mentioned.

In this note, we construct a $(39,\{5,6,7\},1)$-PBD. Thus we have a finite generating set for the PBD-closed set $N_5^{\mathrm{\infty }}$ with at most three inessential elements, where $N_5^{\mathrm{\infty }}=\{1\}\cup \{v:v\ge 5\}$.

In this paper, we prove the NP-hardness of the bottleneck graph bipartition problem and study the complexity status of possible approximation algorithms for some related problems.

This paper concerns neighbour designs in which the elements of each block are arranged on the circumference of a circle. Most of the designs considered comprise a general class of balanced Ouchterlony neighbour designs, which include the balanced circuit designs of Rosa and Huang $[30]$, the neighbour designs of Rees $[29]$, and the more general neighbour designs of Hwang and Lin $[13]$. The class of Rees neighbour designs includes schemes given in 1892 by Lucas $[22]$ for round dances. Isomorphism, species, and adjugate set are defined for balanced Ouchterlony neighbour designs, and some seemingly new methods of constructing such designs are presented. A new class of quasi Rees neighbour designs is defined to cover a situation where Rees neighbour designs cannot exist but where a next best thing may be needed by experimental scientists. Even-handed quasi Rees neighbour designs and even-handed balanced Ouchterlony neighbour designs are defined too, the latter being closely related to serially balanced sequences. This paper does not provide a complete survey of known results, but aims to give the flavour of the subject and to indicate many openings for further research.

A dependence system on a set $S$ is defined by an operator $f$, a function on the power set of $S$ which is extensive ($A$ is included in $f(A)$) and monotone (if $A$ is included in $B$, then $f(A)$ is included in $f(B)$). In this paper, the structure of the set $F(S)$ of all dependence systems on a given set $S$ is studied. The partially ordered set of operators ($f<g$ if for every set $A$, $f(A)$ is included in $g(A)$) is a bounded, complete, completely distributive, atomic, and dual atomic lattice with an involution. It is shown that every operator is a join of transitive operators (usually called closure operators, operators which are idempotent $ff=f$). The study of the class of transitive operators that join-generate all operators makes it possible to express $F(n)$ (the cardinality of the set $F(S)$ of all operators on a set $S$ with $n$ elements) by the Dedekind number $D(n)$. The formula has interesting consequences for dependence system theory.

Let $p(k)$ ($q(k)$) be the smallest number such that in the projective plane, every simple arrangement of $n\ge p(k)$ ($\ge q(k)$) straight lines (pseudolines) contains $k$ lines which determine a $k$-gonal region. The values $p(6)=q(6)=9$ are determined and the existence of $q(k)(\ge p(k))$ is proved.

We introduce a complex version of Golay sequences and show how these may be applied to obtain new Hadamard matrices and complex Hadamard matrices. We also show how to modify the Goethals-Seidel array so that it can be used with complex sequences.

In this paper, we improve the best known algorithm on symmetric equivalence of Hadamard matrices by considering the eigenvalues of symmetric Hadamard matrices. As a byproduct, the eigenvalues of skew Hadamard matrices are also discussed.

The spectrum for $k$-perfect $3k$-cycle systems is considered here for arbitrary $k\not\equiv 0(\mathrm{mod}3)$. Previously, the spectrum when $k=2$ was dealt with by Lindner, Phelps, and Rodger, and that for $k=3$ by the current authors. Here, when $k\equiv 1$ or $5(\mathrm{mod}6)$ and $6k+1$ is prime, we show that the spectrum for $k$-perfect $3k$-cycle systems includes all positive integers congruent to $1(\mathrm{mod}6k)$ (except possibly the isolated case $12k+1$). We also complete the spectrum for $k=4$ and $5$ (except possibly for one isolated case when $k=5$), and deal with other specific small values of $k$.