We give some identities involving sums of powers of the partial sum of q-binomial coefficients, which are q-analogues of Hirschhorn’s identities [Discrete Math. 159 (1996), 273–278] and Zhang’s identity [Discrete Math. 196 (1999), 291–298].

We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain. The combinatorial identity

In 2002 Z. W. Sun published a curious identity involving binomial coefficients. In this paper we present the following generalization of the identity: (x + (m + 1)z) m n=0 (−1) n x + y + nz m − n y + n(z + 1) n =z 0lnm (−1) n n l x + l m − n (1 + z) n+l (1 − z) n−l + (x − m) x m .

This paper proves an identity between flagged Schur polynomials, giving a duality row flags and column flags. generalises both the binomial determinant theorem due to Gessel Viennot symmetric function Aitken. As corollaries we obtain lifts of \(q\)-binomial coefficients polynomials. Our method is path counting argument on novel lattice generalising that used by Viennot.

By counting flags in finite vector spaces, we obtain a q-multinomial analog of a recursion for q-binomial coefficients proved by Nijenhuis, Solow, and Wilf. We use the identity to give a combinatorial proof of a known recurrence for the generalized Galois numbers.