In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

The determinantal conjecture of M. Marcus and G. N. de Oliveira is known in many special cases. The case of 3 × 3 matrices was settled by N. Bebiano, J. K. Merikoski and J. da Providência. The case n = 4 remains open. In this article a technical conjecture is established implying a weakened form of the determinantal conjecture for n = 4.

We answer a question in [10], showing the regular determinantal complexity of the determinant detm is O(m ). We answer questions in, and generalize results of [2], showing there is no rank one determinantal expression for perm m or detm when m ≥ 3. Finally we state and prove several “folklore” results relating different models of computation.

We introduce a new determinantal method to count cycle systems in a directed graph that generalizes Gessel and Viennot’s determinantal method on path systems. The method gives new insight into the enumeration of domino tilings of Aztec diamonds, Aztec pillows, and related regions.

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially. We also present a β-extension of the theorem and a simple determinantal expression for the irreducible characters of the symmetric group.

There is no doubt that the Riemann zeta function is a function of great significance in number theory because of its connection to the distribution of prime numbers. The Ihara zeta function was first introduced by Ihara [5]. This zeta function is associated with a finite graph and aperiodic closed walks in it. Bass’ identity [1] shows that the Ihara zeta function is a determinant. It is shown i...

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen-Li-Louck ...