We study different problems of enumeration of standard paths in the poset of compositions of integers. We show that several problems similar to those considered in the poset of partitions of integers become simpler in this context. We give explicit formulas for generating functions of standard paths in this poset and interesting subposets, and a closed formula for the number of standard paths e...

The standard poset transit function of a poset P is a function TP that assigns to a pair of comparable elements the interval between them, while TP (x, y) = {x, y} for a pair x, y of incomparable elements. Posets in which the standard poset transit function coincides with the shortest-path transit function of its cover-incomparability graph are characterized in three ways, in particular with fo...

Term occurrences of any clause C are determined by their positions. The set of all term partitions defined on subsets of term occurrences of C form a partially ordered set. This poset is isomorphic to the set of all generalizations of C. The structure of this poset can be inferred from the term occurrences in C alone. We can apply these constructions in this poset in machine learning.

A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists polytope whose face lattice poset. In 1990s, Kapranov defined poset as hybrid between of permutohedron that an associahedron, he asked this realizable. Shortly after his question was posed, Reiner Ziegler provided realization. Based on our previous work ne...

We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are downset lattices of finite posets of dimension at most two, and determine a realizer through a recursive relation between weights on the poset associated to...

We outline a method for practical use of an interactive system (APL) to compute the homology of finite partially ordered sets. 1. Prerequisites. All partially ordered sets (posets) are assumed finite. Given a poset , we say that b covers a if b > a and a g c ^ h implies a = c or b = c. Since we deal with finite posets, the order relation can be obtained as the reflexive, transitive clos...

We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable 2-dimensional simplicial complex contains a nonshellable induced subcomplex with less than 8 vertices. We also establish CL-shellability of interval orders and as a consequence obtain a fo...