Real Zeros and Partitions without singleton blocks

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks.Comment: 16 page

Homology of Partial Partitions Ordered by Inclusion

We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find its Betti numbers and prove that there is a basis of its homology that consists of boundaries of combinatorial cross-polytopes of various dimensions

Polynomials with real zeros via special polynomials

In this paper, we use particular polynomials to establish some results on the real rootedness of polynomials. The considered polynomials are Bell polynomials and Hermite polynomials

Polynomials with real zeros via special polynomials

In this paper, we use particular polynomials to establish some results on the real rootedness of polynomials. The considered polynomials are Bell polynomials and Hermite polynomials