VLSI design is probably the most fascinating application area of combinatorial optimization. Virtually all classical combinatorial optimization problems, and many new ones, occur naturally as subtasks. Due to the rapid technological development and major theoretical advances the mathematics of VLSI design has changed significantly over the last ten to twenty years. This survey paper gives an up...

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivit...

We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D̃(n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑ 1≤k...

The practise of theoretical design has not traditionally been one of those areas of applied mathematics where well-posed questions lead directly to well-de ned answers. Instead, combinatorial arrangements are typically rst proposed as designs on more or less obscure and intuitive grounds; various precisely de ned notions of design optimality might be employed, but almost exclusively to con rm t...

We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many cases of combinatorial interest for particular choices of the six parameters that define it. We give a complete classification of the partial differential equa...

By using generating functions, which contained alternative forms, we derive identities, some new and old, for the Bernstein basis functions. Integrating these identities, we also derive combinatorial sums involving binomial coefficients. We give relations between these combinatorial sums and generating functions for special numbers, we investigate relations related to finite sums of the powers ...

The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which relying on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central rôle is played by combinatorial topology, often used to recover the space-time manifold from the other structures involved. An extremely attractive possibility...

We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t = 1 of the Haglund–Haiman–Loehr formula. Also, we pr...

In 1928 Emanuel Sperner presented a simple. Cohen, On the Sperner lemma, J. Theory 2 1967.Abstract. This paper presents elementary combinatorial proofs of Sperners. Advanced topics like degree theory or homology. Before beginning, it.ple combinatorial result known as Sperners lemma. As motivation, we examine a special case of Sperners lemma. Vick, Homology Theory.extremal two-part Sperner famil...

McMahon’s result that states the length and major index statistics are equidistributed on the symmetric group Sn has generalizations to other Coxeter groups. Adin, Brenti and Roichman have defined analogues of those statistics for the hyperoctahedral group, Bn and proved equidistribution theorems. Using combinatorial interpretations of Regev and Roichman’s statistics length and delent, we consi...