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combinat

  

eulerian1

  

first order Eulerian numbers

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

eulerian1(n, k)

Parameters

n, k

-

non-negative integers

Description

• 

The eulerian1(n, k) command counts the number of permutations of pi1pi2...pin of 1&comma;2&comma;...&comma;n that have k ascents, namely, k places where pij<pij+1.

• 

This function can be computed via the recurrence

eulerian1n&comma;k=k+1eulerian1n1&comma;k+nkeulerian1n1&comma;k1

• 

It also satisfies the following identities:

xn=k=0neulerian1n&comma;kx+kn

m!Stirling2n&comma;m=k=0neulerian1n&comma;kknm

  

eulerian1n&comma;k = m=0k−1mn+1mk+1mn = m=0n−1nmkm!nmkStirling2n&comma;m

Examples

withcombinat&colon;

Matrixseqseqeulerian1n&comma;k&comma;k&equals;0..5&comma;n&equals;0..5

10000010000011000014100011111100126662610

(1)

References

  

R.L. Graham, D.E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, Mass., 1989.

See Also

binomial

combinat

combinat[eulerian2]

euler

Stirling2