Sequences, q-Multinomial Identities, Integer Partitions with Kinds, and Generalized Galois Numbers

TL;DR

This paper extends classical binomial and multinomial identities to their q-analog forms using combinatorial proofs involving sequences and statistics, and links integer partitions with kinds to Galois numbers.

Contribution

It introduces new q-analog identities for binomial and multinomial coefficients and connects integer partitions with kinds to Galois number coefficients through combinatorial methods.

Findings

Extended binomial and multinomial identities to q-analog forms

Established a connection between integer partitions with kinds and Galois numbers

Provided combinatorial proofs using sequence statistics

Abstract

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.