We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to p...

Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated re nements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or M obius inversion) are applicable. Unfor...

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices They are useful for coset weight enumeration and weight enumeration of children We determine them in most interesting cases in length at most and in some cases in length We use them to construct group divisible designs packing designs covering designs and t r designs in the sense of Calderbank Delsarte A major ...

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFA’s) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFA’s but, t...

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the prob...

The exact enumeration of most interesting combinatorial problems has exponential computational complexity. The finite-lattice method reduces this complexity for most two-dimensional problems. The basic idea is to enumerate the problem on small finite lattices using a transfer-matrix formalism. Systematically grow the size of the lattices and combine the results in order to obtain the desired se...

Local rank modulation scheme was suggested recently for representing information in flash memories in order to overcome drawbacks of rank modulation. For 0 < s ≤ t ≤ n with s divides n, an (s, t, n)-LRM scheme is a local rank modulation scheme where the n cells are locally viewed cyclically through a sliding window of size t resulting in a sequence of small permutations which requires less comp...

We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound τ >...

One of the most attractive problems for post-quantum secure cryptographic schemes is the LWE problem. Beside combinatorial and algebraic attacks, LWE can be solved by a lattice-based Bounded Distance Decoding (BDD) approach. We provide the first parallel implementation of an enumeration-based BDD algorithm that employs the Lindner-Peikert and Linear Length pruning strategies. We ran our algorit...

Constraint logic programming over finite integer domains allows a declarative and flexible statement of combinatorial optimization problems. The paradigm used is "constraint and generate". Constraints prune in advance the search space, and then, a enumeration phase, also called labeling, and a search strategy are needed to find the optimal solution. In this paper we introduce the integration of...