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 provide a formula for the number of edges of the Hasse diagram of the
independent subsets of the h-th power of a path ordered by inclusion. For h=1
such a value is the number of edges of a Fibonacci cube. We show that, in
general, the number of edges of the diagram is obtained by convolution of a
Fibonacci-like sequence with itself.Comment: Preprint submitted to Electronic Notes in Discrete Mathematic
In this paper, we investigate the notion of partition of a finite partially
ordered set (poset, for short). We will define three different notions of
partition of a poset, namely, monotone, regular, and open partition. For each
of these notions we will find three equivalent definitions, that will be shown
to be equivalent. We start by defining partitions of a poset in terms of fibres
of some surjection having the poset as domain. We then obtain combinatorial
characterisations of such notions in terms of blocks, without reference to
surjection. Finally, we give a further, equivalent definition of each kind of
partition by means of analogues of equivalence relations.Comment: Preprint submitted for publication in the book "From Combinatorics to
Philosophy: The Legacy of G.-C. Rota
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.