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combinat

  

eulerian2

  

second order Eulerian numbers

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

eulerian2(n, k)

Parameters

n, k

-

non-negative integers

Description

• 

The eulerian2(n, k) command counts the number of permutations pi1pi2`...`pi2n of the multi-set 1,1,2,2,...,n,n having two properties:

1. 

All numbers between the two occurrences of m are greater than m.

2. 

There are k ascents, namely, k places where pij<pij+1.

• 

This function can be computed via the recurrence

eulerian2n&comma;k&equals;k&plus;1eulerian2n1&comma;k&plus;2nk1eulerian2n1&comma;k1

• 

Second-order Eulerian numbers are important because of their connection with Stirling numbers. For integers m and 0n we have:

Stirling2m&comma;mn&equals;k&equals;0neulerian2n&comma;kbinomialm&plus;n1k&comma;2n

  

Stirling1m&comma;mn = 1nStirling1m&comma;mn = k&equals;0neulerian2n&comma;kbinomialm&plus;k&comma;2n

Examples

withcombinat&colon;

Matrixseqseqeulerian2n&comma;k&comma;k&equals;0..5&comma;n&equals;0..5

1000001000001200001860001225824001523284441200

(1)

References

  

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

See Also

combinat

combinat[eulerian1]

euler

Stirling1

Stirling2

 


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