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bibkey: han2000scott authors: Guo-Niu Han year: 2000 title: “Généralisation de l’identité de Scott sur les permanents” doi: 10.1016/S0024-3795(00)00035-5 url: https://doi.org/10.1016/S0024-3795(00)00035-5 claim: “The cycle-geodesic product is a specialization of the generalized Scott permanent identity.” strata_touched:

• D5/S3/Combinatorics/Permanental/CycleGeodesic/CycleGeodesicProduct license: citation-only triage: anchor

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DOI: 10.1016/S0024-3795(00)00035-5

Source: https://doi.org/10.1016/S0024-3795(00)00035-5

Identity and specialization

Linear Algebra and its Applications 311, pages 25–34. Han–Krattenthaler’s exposition, arXiv:math/0003072v3, Section 5, prints the specialization

$$\mathrm{PER}(x^n-1,y^n+a+b)=(-1{)}^{n+1}\frac{{\prod }_{i=1}^n(i-(n-i)(a+b))}{(a+b+1{)}^n}.$$

Putting $a+b=-q$ and including the cycle-geodesic row factors yields

$$\mathrm{perm}{\gamma }_n(t)=n^{-n}\prod _{k=0}^{n-1}(n-k+kq),q=e^{2\pi it}.$$

This exact product is literature-attested. Its Lean proof uses a Cauchy permanent expressed as a Gaudin determinant and evaluates the latter on a weighted Vandermonde basis. The literature identity is not an axiom of that proof.

Related source: https://arxiv.org/abs/math/0003072v3. Lascoux’s discussion at https://arxiv.org/abs/1002.3785 attributes this case to Han.