This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, ...

We present a general result that, using the theory of symmetric functions, produces several new classes of symmetric unimodal polynomials. The result has applications to enumerative combinatorics including the proof of a conjecture by R. Stanley.

Enumerative combinatorics is about counting. The typical question is to find the number of objects with a given set of properties. However, enumerative combinatorics is not just about counting. In “real life”, when we talk about counting, we imagine lining up a set of objects and counting them off: 1, 2, 3, . . .. However, families of combinatorial objects do not come to us in a natural linear ...

In this scheme, all tasks of enumerative combinatorics are solved, which include the construction a procedure for listing its outcomes, determining their number, finding probability distribution, and solving numbering task in direct reverse formulations. It is suggested that results used as basis universal modeling outcomes.

A holonomic (i.e., D-finite, or P -recursive) sequence is one that satisfies a linear recursion relation with polynomial coefficients. A multisum sequence is one that is given by a multisum of a proper hypergeometric term. A fundamental theorem of Wilf-Zeilberger states that every multisum sequence is holonomic. For over 15 years, it was accepted as a reasonable conjecture that the converse hol...

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have...

The notion of a descent polynomial, function in enumerative combinatorics that counts permutations with specific properties, enjoys revived recent research interest due to its connection other important notions combinatorics, viz. peak polynomials and symmetric functions. We define the dm(I,n) as generalization polynomial we prove for any positive integer m, this is n sufficiently large (simila...