Let α be a formal variable and Fw be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form Fw = E(F c w), where E and F c w respectively denote the species of sets and of connected Fw-structures. Multiplying by α the weight of each F c wstructure yields the species Fw(α) = E(F c αw). We introduce a “universal” virtual weighted species, Λ,...

The fundamental problem of enumerative combinatorics is to determine the number of elements of a set. More precisely, given an infinite indexed collection {Ai}i∈I of finite sets, we want to find a formula for the cardinality of Ai as a function of i, or at least a method for determining the number of elements of Ai that is easier (or more interesting) than counting them one at a time. There are...

Abstract: This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on specie...

A permutation π contains a permutation σ as a pattern if it contains a subsequence of length |σ| whose elements are in the same relative order as in the permutation σ. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption NP 6⊆ co-NP/poly) by introducing a new polynomial reduction...

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniq...

In this work we introduce the triangular arrays of depth greater than 1 given by linear recurrences, that generalize some well-known recurrences that appear in enumerative combinatorics. In particular, we focussed on triangular arrays of depth 2, since they are closely related to the solution of linear three–term recurrences. We show through some simple examples how these triangular arrays appe...

We investigate properties of a multivariate function E(m1,m2, . . . ,mr), called orbicyclic, that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. E(m1,m2, . . . ,mr) proves to be multiplicative, and a simple formula for its calculation is provided. It is shown that the necessary and sufficient conditions for this function to vanish are equivalent to f...

A distributive lattice L with 0 is finitary if every interval is finite. A function f : N0 ! N0 is a cover function for L if every element with n lower covers has f ðnÞ upper covers. All non-decreasing cover functions have been characterized by the author ([2]), settling a 1975 conjecture of Richard P. Stanley. In this paper, all finitary distributive lattices with cover functions are character...

Motivated by applications in enumerative combinatorics and the analysis of algorithms we investigate the number of gaps and the length of the longest gap in a discrete random sample from a general distribution. We obtain necessary and sufficient conditions on the underlying distribution for the gaps to vanish asymptotically (with probability 1, or in probability), and we study the limiting dist...