A generating function framework for the no-feedback card guessing game after riffle shuffles

Abstract

We introduce a generating-function framework for analyzing the no-feedback card-guessing game after k Gilbert–Shannon–Reeds riffle shuffles. We show that the distribution of the card appearing in position i can be expressed as a structured mixture of 2^k tractable components, each corresponding to a sum of independent Bernoulli trials. From this decomposition, we derive an explicit closed-form expression for the probability generating function, represented as a product of binomial-type polynomials with a clear and systematic structure, valid for any number of cards n and any number of shuffles k. This formulation replaces recursive convolutions with a single analytic expression, enabling efficient computation and revealing the combinatorial–probabilistic structure underlying riffle shuffles. Beyond exact evaluation, the framework connects optimal no-feedback strategies with the generating functions and suggests asymptotic behavior in both the fixed-k, large-n and fixed-n, large-k regimes.