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Weighted Inversion Statistic


A statistic w on the symmetric group S_n is called a weighted inversion statistic if there exists an upper triangular matrix W=(w_(ij)) such that

 w(sigma)=sum_(i<j)chi(sigma_i>sigma_j)w_(ij),

where chi is the characteristic function.

The inversion count (w_(ij)=1 for i<j) defined by Cramer (1750) and the major index (w_(i,i+1)=i; w_(ij)=0 otherwise) defined by MacMahon (1913) are both weighted inversion statistics (Degenhardt and Milne).


See also

Inversion Statistic, Symmetric Group

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References

Cramer, G. "Intr. à l'analyse de lignes courbes algébriques." Geneva, 657-659, 1750.Degenhardt, S. L. and Milne, S. C. "Weighted Inversion Statistics and Their Symmetry Group." J. Combin. Th. Ser. A. 90, 49-103, 2000.MacMahon, P. A. "The Indices of Permutations." Amer. J. Math. 35, 281-322, 1913.

Referenced on Wolfram|Alpha

Weighted Inversion Statistic

Cite this as:

Weisstein, Eric W. "Weighted Inversion Statistic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeightedInversionStatistic.html

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