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Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences
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Huilan Li

Department of Mathematics

Drexel University

3141 Chestnut Street

Philadelphia, PA 19104

USA

Trueman MacHenry

Department of Mathematics and Statistics

York University

Toronto, ON M3J 1P3

Canada

**Abstract:**

In this paper we construct two types of Hessenberg matrices with the
property 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 in the elementary symmetric polynomial
basis. Among them are the generalized Fibonacci polynomials and the
generalized Lucas polynomials, which already have these sweeping
representation properties. Among the integer sequences discussed are
the Chebyshev 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.

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(Concerned with sequences
A000005
A000010
A000108
A000129
A000203
A000217
A001550
A001608
A001700
A001973
A007434
A008275
A008277
A011973
A015441
A071951
A078512
A113501.)

Received July 1 2012;
revised versions received August 5 2012; October 27 2012;
January 24 2013; February 17 2013.
Published in *Journal of Integer Sequences*, March 2 2013.

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