Proof of Scott’s conjecture
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- Proc. Amer. Math. Soc. 87 (1983), 203-207 Request permission
Abstract:
We give a proof of Conjecture 7 in [2, p. 155] first stated in 1881 by R. F. Scott [4]. It reads as follows: Conjecture 7 (R. F. Scott). Let ${x_1}, \ldots ,{x_n}$ and ${y_1}, \ldots ,{y_n}$ be the distinct roots of ${x^n} - 1 = 0$ and ${y^n} + 1 = 0$, respectively. Let $A$ be the $n \times n$ matrix whose $(i,j)$ entry is $1/({x_i} - {y_i}),i,j = 1, \ldots ,n$. Then \[ \left | {\operatorname {per} (A)} \right | = \left \{ {\begin {array}{*{20}{c}} {n{{(1 \cdot 3 \cdot 5 \cdots (n - 2))}^2}/{2^{n}}}, & {{\text {if}}\;n\;{\text {is }}\operatorname {odd} ,} \\ {0,} & {{\text {if}}\;n\;{\text {is}}\;\operatorname {even} {\text {.}}} \\ \end {array} } \right .\] Actually, our proof gives more, namely an explicit expression for ${\operatorname {per}}(A)$ (see Theorem 2.1).References
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C. W. Borchardt, Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Funktion, Monatsb. Akad. Wiss. Berlin 1888 (1855), 165-171; or Crelle’s J. 53 (1855), 193-198; or Gesammelte Werke, 97-105.
- Henryk Minc, Permanents, Encyclopedia of Mathematics and its Applications, vol. 6, Addison-Wesley Publishing Co., Reading, Mass., 1978. With a foreword by Marvin Marcus. MR 504978
- Thomas Muir, A treatise on the theory of determinants, Dover Publications, Inc., New York, 1960. Revised and enlarged by William H. Metzler. MR 0114826 R. F. Scott, Mathematical notes, Messenger of Math. 10 (1881), 142-149.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 203-207
- MSC: Primary 15A15; Secondary 12D99
- DOI: https://doi.org/10.1090/S0002-9939-1983-0681822-9
- MathSciNet review: 681822