Zero-Run Spectra of the $(3,2)$ Raney numbers Modulo Primes

Abstract

We study the zero-run structure of the $(3,2)$-Raney numbers modulo a prime $p$. For every prime $p$, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in $0\le n<p^m$. Interestingly, these results fall into three cases: $p=2$, $p=3$, and $p\geq5$, and the behaviors in these three cases are very different. For $p=2$, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For $p=3$, we characterize exactly the nonzero terms and determine their residues. For $p\ge 5$, the zero runs are governed by a multiscale system of residue intervals modulo powers of $p$, from which both the record values and the complete zero-run spectrum are obtained.