The q-Log-Concavity and Unimodality of q-Kaplansky Numbers

TL;DR

This paper establishes the unimodality and strong q-log-concavity of q-Kaplansky numbers, linking them to Gaussian polynomials and permutation statistics, thus advancing understanding of their combinatorial properties.

Contribution

It proves the unimodality and q-log-concavity of q-Kaplansky numbers and connects them to Gaussian polynomials and permutation statistics, providing new insights into their structure.

Findings

Proved the unimodality of q-Kaplansky numbers.

Established the strong q-log-concavity of q-Kaplansky numbers.

Connected q-Kaplansky numbers to Gaussian polynomials and permutation statistics.

Abstract

$q$-Kaplansky numbers were considered by Chen and Rota. We find that $q$-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on the work of Reiner and Stanton, we establish the unimodality of $q$-Kaplansky numbers. We also show that $q$-Kaplansky numbers are the generating functions for the inversion number and the major index of two special kinds of $(0,1)$-sequences. Furthermore, we show that $q$-Kaplansky numbers are strongly $q$-log-concave.