Journal of Integer Sequences, Vol. 16 (2013), Article 13.3.5

Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences

Huilan Li
Department of Mathematics
Drexel University
3141 Chestnut Street
Philadelphia, PA 19104

Trueman MacHenry
Department of Mathematics and Statistics
York University
Toronto, ON M3J 1P3


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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