Permanents, Determinants, Weighted Isobaric Polynomials and Integer Sequences

TL;DR

This paper constructs Hessenberg matrices whose determinants and permanents generate all weighted isobaric polynomials, enabling representation of all linearly recurrent integer sequences, including classical polynomials and number sequences.

Contribution

It introduces two types of Hessenberg matrices that represent weighted isobaric polynomials as determinants and permanents, expanding the understanding of their algebraic and combinatorial properties.

Findings

Weighted isobaric polynomials appear as determinants and permanents of specific Hessenberg matrices.

All linearly recurrent integer sequences can be represented via evaluations of WIPs.

Classical sequences like Fibonacci, Lucas, Chebyshev, Stirling, and Catalan are included.

Abstract

In this paper we construct two types of Hessenberg matrices with the properties that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is linearly recurrent is representable by (an evaluation of) some linearly recurrent sequence of WIPs. WIPs are symmetric polynomials written on the elementary symmetric polynomial basis. Among them are the generalized Fibonacci polynomials and the generalized Lucas polynomials, which already have these sweeping representing properties. Among the integer sequences discussed are the Chebychev polynomials of the 2nd kind, the Stirling numbers of the 1st and 2nd kind, the Catalan numbers, and the triangular numbers, as well as all sequences which are either multiplicative arithmetic functions or additive arithmetic functions.