We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom (α → β) ∨ (β → α) = 1. (Since Gödel algebras are locally finite, 'finitely generated', 'finitely presented', and 'finite' have identical meaning in this paper.) We achieve this result using ordered partitions of finite sets as a key tool to investigate the category opposite to finitely generated Gödel algebras (forests and open order-preserving maps). We give two applications of our main result. We prove that finitely presented Gödel algebras have free products with amalgamation; and we easily obtain a recursive formula for the cardinality of the free Gödel algebra over a finite number of generators first established by A. Horn.

We derive explicit formulas for the connection constants between sequences of polynomials in terms of their roots. As an application, we give some properties about the inversion of triangular matrices. By means of the concept of punctured partition of an integer, here introduced to compute connection constants, we also obtain a new expression of the relationship between elementary and complete homogeneous symmetric functions.

Abstract. In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that appears to be relevant to the so-called Boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colourings and another one in terms of certain labelled Eulerian digraphs.

We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of Bn, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.We conclude with other applications of this set encoding: it gives the relationship between the elementary and complete symmetric functions as well as a new formula for the Bell numbers. 0