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Worpitzky's Identity

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Worpitzky's identity expresses a power as a sum involving Eulerian numbers and binomial coefficients,

x^n=sum_(k=0)^n<n; k>(x+k; n),

where <n; k> is an Eulerian number and (n; k) is a binomial coefficient (Worpitzky 1883; Comtet 1974, p. 242). Here <n; k> counts permutations of n elements with k permutation ascents. With the alternative convention A(n,k)=<n; k-1>, the same identity is

x^n=sum_(k=1)^nA(n,k)(x+k-1; n).

These forms hold for n>=1 and agree after shifting the summation index.

See also

Binomial Sums, Eulerian Number

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References

Buhler, J. and Graham, R. "Juggling Drops and Descents." Amer. Math. Monthly 101, 507-519, 1994.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.Graham, R. L.; Knuth, D. E.; and Patashnik, O. §6.2 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Stanton, D. Constructive Combinatorics. New York: Springer-Verlag, 1986.Worpitzky, J. "Studien über die Bernoullischen und Eulerischen Zahlen." J. reine angew. Math. 94, 203-232, 1883.

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Worpitzky's Identity

Cite this as:

Weisstein, Eric W. "Worpitzky's Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WorpitzkysIdentity.html

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