  Dicke states are completely symmetric states of multiple qubits (2-level
systems), and qudit Dicke states are their $d$-level generalization. We define
here $q$-deformed qudit Dicke states using the quantum algebra $su_q(d)$. We
show that these states can be compactly expressed as a weighted sum over
permutations with $q$-factors involving the so-called inversion number, an
important permutation statistic in Combinatorics. We use this result to compute
the bipartite entanglement entropy of these states. We also discuss the
preparation of these states on a quantum computer, and show that introducing a
$q$-dependence does not change the circuit gate count.
