The exhaustive generation of combinatorial objects has a vast range of practical applications and is a common theme in the combinatorial research field. But most previous works in this area concentrate in the efficient generation of particular families of combinatorial objects. The novel approach of the work presented here is to provide efficient generic algorithms, where the input is not just ...

The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects “Young books” are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases...

We deal with the enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings and ice models. The notion of an α-orientation unifies many different combinatorial structures, including the afore mentioned. We ask for the number of α-orientations and also for special instances thereof, such as Schny...

We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per component of the forest. Equivalently, we count regular maps equipped with a spanning tree, with a weight z per face and a weight μ := u+ 1 per internally active edge, in the sense of Tutte. This enumeration problem corresponds to the limit q → 0 of the q-state Potts m...

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α : V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations α-orien...

In the plane Z× Z a cell is a unit square and a polyomino is a finite connected union of cells. Polyominoes are defined up to translations. Since they have been introduced by Golomb [20], polyominoes have become quite popular combinatorial objects and have shown relations with many mathematical problems, such as tilings [6], or games [19] among many others. Two of the most relevant combinatoria...

The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry [4] in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given [1, 2, 3, 5], but these proofs generally ...

We give recurrence relations for the enumeration of symmetric elements within four classes arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These are motivated by study RNA secondary structures. For example, classic structures correspond 3412-avoiding with adjacent transpositions, base triples may be represented as crossings. Our...

Abstract We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by basic combinator ${{\mathbf{M}}}$ . This combinator, named as Mockingbird Smullyan, is defined rewrite rule ${{\mathbf{M}}} \mathsf{x}_1 \to \mathsf{x}_1$ prove that reflexive transitive closure this relation a partial on terms all connected components its graph are Hasse dia...

Pairs of graded graphs, together with the Fomin property graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is one Young integer partitions, allowing us to connect number standard tableaux and numbers permutations. Here, we use operads, that algebraic devices abstracting notion composition objects, build pairs graphs. For...