Abstract. We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specialization of this identity. Combinatorial or probabilistic interpr...

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form n = ∑L−1 j=1 jfj, with f1 ≤ i−1, fL−1 ≤ i ′−1 and fj+fj+1 ≤ k. The bosonic side of the identities involves q-deformations of the coefficients of xa in the expansion of (1 + x + · · · + xk)L. A combinatorial interpretation for these q-multinomial coefficients is given using Durfe...

It is well known that a system of power polynomial equations can be reduced to a single-variable polynomial equation by exploiting the so-called Newton’s identities. In this work, by further exploring Newton’s identities, we discover a binomial decomposition rule for composite elementary symmetric polynomials. Utilizing this decomposition rule, we solve three types of systems of composite power...

The focus of this paper is to study the HOMFLY polynomial of (2, n)-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of (2, n)-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental ...

The Martin polynomial of an oriented Eulerian graph encodes information about families of cycles in the graph. This paper uses a transformation of the Martin polynomial that facilitates standard combinatorial manipulations. These manipulations result in several new identities for the Martin polynomial, including a di.erentiation formula. These identities are then applied to get new combinatoria...

We consider a certain decomposition of the matrix algebra Mn(F ), where F is a field. The commutation relations of that decomposition yield an n × n matrix Mn , which determines the multilinear polynomial identities of Mn(F ). Thus if char(F ) = 0, the matrix Mn ) determines the polynomial identities of Mn(F ). We show that M Mn(F ) is very close to the tensor product of two n × n Vandermonde m...