first order Eulerian numbers - Maple Help

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combinat[eulerian1] - first order Eulerian numbers

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&equals;k&plus;1eulerian1n1&comma;k&plus;nkeulerian1n1&comma;k1

• 

It also satisfies the following identities:

xn&equals;k&equals;0neulerian1n&comma;kbinomialx&plus;k&comma;n

m&excl;Stirling2n&comma;m&equals;k&equals;0neulerian1n&comma;kbinomialk&comma;nm

  

eulerian1n&comma;k = m&equals;0k1mbinomialn&plus;1&comma;mk&plus;1mn = m&equals;0n1nmkm&excl;binomialnm&comma;kStirling2n&comma;m

Examples

withcombinat&colon;

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

10000010000011000014100011111100126662610

(1)

See Also

combinat, combinat[eulerian2], euler

References

  

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


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