In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström–Gessel–Viennotmethod and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides “graphical proofs” for classical determinantal identities like the Cauchy-Binet formula, Dodgson’s condensation fo...

In this paper we will present a new method to calculate determinants of square matrices. The method is based on the Chio-Dodgson's condensation formula and our approach automatically affects in reducing the order of determinants by two. Also, using the Chio's condensation method we present an inductive proof of Dodgson's determinantal identity.

Sylvester’s identity is a classical determinantal identity with a straightforward linear algebra proof. We present combinatorial proofs of several non-commutative extensions, and find a β-extension that is both a generalization of Sylvester’s identity and the β-extension of the quantum MacMahon master theorem.

We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for sλ(x/y). This new expression gives rise to a determinantal formula for sλ(x/y). I...

Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a β-extension that is both a generalization of Sylvester's identity and the β-extension of the MacMahon master theorem.

Our recent paper [5] provides proofs of certain generalizations of two classical determinantal identities, one by Bressoud and Wei [1] and one by Koike [8]. Both of these identities are extensions of the Jacobi-Trudi identity, an identity that provides a determinantal representation of the Schur function. Here we provide lattice path proofs of these generalized idetities. We give the barest of ...

We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz’s identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg’s Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.