Enumeration of multiplex juggling card sequences using generalized q-derivatives
Yumin Cho, Jaehyun Kim, Jang Soo Kim, Nakyung Lee
TL;DR
This work resolves the enumeration of multiplex juggling card sequences by linking cards to $(b,k)$-embeddings and introducing a generalized $q$-derivative operator to handle card sequences. It proves Butler et al.'s conjecture for capacity $k=2$ and provides explicit generating functions for general capacity $k$ and length $\ell$, culminating in a rational generating function in $x$ for fixed $(k,\ell)$. The approach blends combinatorial embeddings, symmetric-function techniques, and a multi-variable derivative operation to obtain compact expressions such as $\sum_{b\ge0} J(b,k,\ell)x^b = [z_1^k \cdots z_\ell^k]\left( \frac{1}{1-z_1} \cdots \frac{1}{1-z_\ell} D_{z_1,z_2} \cdots D_{z_1,\dots,z_\ell} \frac{1}{2 - h_k(1,x,xz_1,\dots,xz_\ell)} \right)$. Consequently, these generating functions are rational in $x$, implying linear recurrences for the sequences $J(b,k,\ell)$ and laying a foundation for multivariate extensions and related open problems on $J_0(b,k,\ell)$ and higher-parameter generating functions.
Abstract
In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card that represents multiplex juggling patterns in a natural bijective way. They conjectured a formula for the generating function for the number of multiplex juggling cards with capacity 2. In this paper we prove their conjecture. More generally, we find an explicit formula for the generating function with any capacity. We also find an expression for the generating function for multiplex juggling card sequences by introducing a generalization of the q-derivative operator. As a consequence, we show that this generating function is a rational function.