We study the notion of a "differential 2-rig", category R with coproducts and monoidal structure distributing over them, also equipped an endofunctor D : -> that satisfies categorified analogue Leibniz rule. This is intended as tool to unify various applications such categories computer science, algebraic topology, enumerative combinatorics. The theory differential 2-rigs has geometric flavour ...

This paper will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occ...

In the study of various objects indexed by permutations, a natural notion of minimal excluded structure, now known as a permutation pattern, has emerged and found diverse applications. One of the earliest results from the study of permutation pattern avoidance in enumerative combinatorics is that the Catalan numbers cn count the permutations of size n that avoid any fixed pattern of size three....

simplicial complex, 122, 136,164acyclic orientation, 5, 218affine hull, 52affine projection, 56alcoved polytope, 227alcoved triangulation, 228alcoves, 227Andrews, George, 125anti-isomorphic, 239anti-symmetric, 11antichain, 13, 28, 182antistar, 168Appel, Kenneth, 2, 20arrangement of hyperplanes, 59, 74, 215ascent, 171, 191big, 235<l...

In this column we review the following books. A combinatorial design is a set of sets of (say) {1,. .. , n} that have various properties, such as that no two of them have a large intersection. For what parameters do such designs exist? This is an interesting question that touches on many branches of math for both origin and application. 2. Combinatorics of Permutations by Miklós Bóna. Review by...

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableau...

Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus generalizing previous results on the more restricted class of codimension-one tilings. In particular this method gives access to counting formulas, which are ...

The purpose of this dissertation is to study certain classes of permutations and plane partitions of truncated shapes. We establish some of their enumerative and combinatorial properties. The proofs develop methods and interpretations within various fields of algebraic combinatorics, most notably the theory of symmetric functions and their combinatorial properties. The first chapter of this the...

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ , where g ≈ 0.00909941 and ρ ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n v...

ArXiv : math.CO/0403017 v 1 1 March 2004 Summary Combinatorial interpretation of the fibonomial coefficients recently attampted by the present author [1,2] and presented here with suitable improvements results in a proposal of a might be combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided within the context of the classical combinat...