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On the spectrum of the perfect matching derangement graph

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Abstract

We prove the alternating sign conjecture for the perfect matching derangement graph.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. The use of the word ‘derangement’ in this context is somewhat awkward, but now standard. One can draw an analogy between the graph \(\Gamma _{2n}\) and the derangement graph, whose vertices are the set of permutations on an n element set, and whose edges are pairs \(\{\pi ,\sigma \}\) of permutations that are derangements of one another, meaning that, in one-line notation, \(\pi \) and \(\sigma \) disagree in every position. In that sense, one could say that \(\pi \) and \(\sigma \) do not intersect. Similarly, if two perfect matchings M and \(M'\) do not intersect, it is said that M and \(M'\) are derangements of one another.

  2. See, e.g., [7], and references therein, as well as the bibliography below.

  3. For a beautiful introduction to this constellation of ideas, see [1].

  4. Actually, Okounkov and Olshanski discuss only Jack polynomials, which they write as \(P_\mu (x;\theta )\). Here, \(\theta =1/\alpha \), where \(\alpha \) is the usual Jack parameter. But Macdonald and Stanley show (see, e.g., [16], Proposition 1.2 or [10], p. 408) that the zonal polynomials are obtained from the Jack polynomials by setting \(\alpha =2\). For this reason, we call \(P_\mu (x;\frac{1}{2})\) zonal polynomials.

  5. Note that \(a'(\square )\) and \(\ell '(\square )\) are computed relative to the origin of \(\mu \). Note also that, to be consistent with the initial condition, we must set \(\langle z\rangle _\varnothing =\psi _\varnothing =1\).

References

  1. Godsil, C., Meagher, K.: Erdős-Ko-Rado Theorems: Algebraic Approaches. Cambridge University Press, Cambridge (2016)

    MATH  Google Scholar 

  2. Godsil, C., Meagher, K.: An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings. Ars. Math. Contemp. 12, 205–217 (2017)

    Article  MathSciNet  Google Scholar 

  3. James, A.T.: Distributions of matrix variables and latent roots derived from normal samples. Ann. Math. Stat. 35, 475–501 (1964)

    Article  Google Scholar 

  4. Knop, F., Sahi, S.: Difference equations and symmetric polynomials defined by their zeros. Internat. Math. Res. Notices 10, 473–486 (1996)

    Article  MathSciNet  Google Scholar 

  5. Ku, C.-Y., Wales, D.: Eigenvalues of the derangement graph. J. Comb. Theory Ser. A 117(3), 289–312 (2010)

    Article  MathSciNet  Google Scholar 

  6. Ku, C.-Y., Wong, K.-B.: Solving the Ku-Wales conjecture on the eigenvalues of the derangement graph. European J. Comb. 34(6), 941–956 (2013)

    Article  MathSciNet  Google Scholar 

  7. Ku, C.-Y., Wong, K.-B.: Eigenvalues of the matching derangement graph. J. Algebraic Combin. 48, 627–646 (2018)

    Article  MathSciNet  Google Scholar 

  8. Ku, C.-Y., Wong, K.-B., Koh, S.: Alternating sign property of the matching derangement graph, preprint of June 9, (2021)

  9. Lindzey, N.: Erdős-Ko-Rado for perfect matchings. European J. Combin. 65, 130–142 (2017)

    Article  MathSciNet  Google Scholar 

  10. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  11. Meagher, K., Moura, L.: Erdős-Ko-Rado theorems for uniform set-partition systems, Electron. J. Combin., 12, (2005), #R40

  12. Okounkov, A.: (Shifted) Macdonald polynomials: \(q\)-integral representation and combinatorial formula. Compositio Math. 112, 147–182 (1998)

    Article  MathSciNet  Google Scholar 

  13. Okounkov, A., Olshanski, G.: Shifted Jack polynomials, binomial formula, and applications. Math. Res. Lett. 4, 69–78 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Renteln, P.: On the spectrum of the derangement graph, Electron. J. Combin., 14 (2007), #R82

  15. Sahi, S.: Interpolation, integrality, and a generalization of Macdonald’s polynomials. Math. Res. Notices 10, 457–471 (1996)

  16. Stanley, R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77, 76–115 (1989)

    Article  MathSciNet  Google Scholar 

  17. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

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  1. Department of Physics, California State University, San Bernardino, CA, 92407, USA

    Paul Renteln

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Renteln, P. On the spectrum of the perfect matching derangement graph. J Algebr Comb 56, 215–228 (2022). https://doi.org/10.1007/s10801-021-01105-y

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