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We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common features: combinatorial interpretations, Pascal-like recurrences, inverse relations with their signed versions, and interpretations as coefficients of change between polynomial bases., Comment: 10 pages

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $56$ points. One new $(56,16,18)$ and many new $(56,16,6)$ designs are discovered, and non-existence of $(56,12,9)$ and $(56,20,19)$ designs with certain automorphism groups is proved. The number of known symmetric $(78,22,6)$ designs is also significantly increased., Comment: 18 pages

We establish the existence of simple designs with parameters $2$-$(55, 10, 4)$, $3$-$(20, 5, 4)$, $3$-$(21, 7, 30)$, $4$-$(15, 5, 2)$, $4$-$(16, 8, 45)$, $5$-$(16, 7, 10)$, and $5$-$(17, 8, 40)$, which have previously been unknown. For the corresponding $t$, $v$, and $k$, we study the set of all $\lambda$ for which simple $t$-$(v, k, \lambda)$ designs exist.

We prove the existence of designs with parameters 4-(18, 9, 56), 4-(18, 9, 70), 4-(19, 9, 84), and 4-(19, 9, 105) by assuming suitable automorphism groups and performing a computer search using the method of Kramer and Mesner.

Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids.

We propose a combination of two known computational methods for the construction of designs with prescribed groups of automorphisms: the Kramer–Mesner method and the method of tactical decompositions. This combined method is used to construct new unitals with parameters 2-(65, 5, 1). © 2011 Wiley Periodicals, Inc. J Combin Designs 19:290-303, 2011

A t-(v, k, lambda) design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and investigate tilings consisting of Singer difference sets.

It is known that the classical unital arising from the Hermitian curve in PG(2,9) does not have a 2-coloring without monochromatic lines. Here we show that for q≥4 the Hermitian curve in PG(2,q2) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital.