The number of terms in the permanent and the determinant of a generic circulant matrix

2003/01/07 by Hugh Thomas, Thomas, Hugh
Computer Science · Mathematics · Physics and Astronomy · #05A15 #05E05 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #math.CO #msc:05A15 #msc:05E05

paper · pdf · doi:10.48550/arxiv.math/0301048

6 pages; 1 figure

arxiv created 2003/01/07 · openalex publication_date 2003/01/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n)=p(n). The proof uses symmetric functions.

Citations

• Brick tabloids and the connection matrices between bases of symmetric functions

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