On a Combinatorial Identity

A combinatorial identity and the finite dual of infinite dihedral group algebra

In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $γ_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite dihedral group.

Combinatorial Identities Related to Representations of $U_q(\tilde{gl_2})$

Recently N.Jing discovered a certain combinatorial identity from validity of the Serre relations in some vertex representations of quantum Kac-Moody algebras. We generalize this identity, in particular, extending it from polynomials to elliptic functions, and interprete the obtained identities in terms of tensor products of evaluation representations of the quantum loop algebra $U_q(\tilde{gl_2})$ or the elliptic quantum group $E_{ρ,γ}(sl_2)$.

置换基 MacDonald 多项式恒等式的组合证明

Combinatorial proof of a permuted basement Macdonald polynomial identity

研究置换基 MacDonald 多项式恒等式，通过将 Concha - Lapointe 恒等式细化到特定子族，给出组合证明，并揭示其与 Kazhdan - Lusztig 对合的等价关系。

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra $A_{n-1}^{(1)}.$ In this first paper, we focus on combinatorial aspects. We give a $n^2$-coloured generalisation of Primc's identity by constructing a $n^2 \times n^2$ matrix of difference conditions, Primc's original identities corresponding to $n=2$ and $n=3$. While most coloured partition identities in the literature connect partitions with difference conditions to partitio…

Codes and shifted codes of partitions

In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernstein's Theorem. We show that this identity is a straightforward consequence of the classical result. We also show how a similar approach using the codes of partitions can be generalized from Schur functions to also include Schur $Q$-functions and derive the combinatorial formulation for both cases. We then apply them by examining the Littlewood-Richardson and Pieri Rules.

A Shuffle Theorem for Paths Under Any Line

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose $x$ and $y$ intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $GL_{l}$ charac…