A010030 - OEIS

A010030

Irregular triangle read by rows: T(n,k) (n >= 1, 0 <= k <= [n/2]) = number of permutations of 1..n with [n/2]-k runs of consecutive pairs up and down (divided by 2).

1, 1, 0, 3, 0, 3, 8, 1, 25, 28, 7, 17, 155, 143, 45, 259, 1005, 933, 323, 131, 2770, 7488, 7150, 2621, 3177, 27978, 64164, 62310, 23811, 1281, 51433, 294602, 619986, 607445, 239653

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OFFSET

1,4

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.

LINKS

Table of n, a(n) for n=1..35.

FORMULA

G.f. for number of permutations of 1..n by number of runs of consecutive pairs up and down is Sum(n!*(((1-y)*(2*x^2-x^3)-x)/((1-y)*x^2-1))^n,n = 0 .. infinity), cf. A010029. - Vladeta Jovovic, Nov 23 2007

EXAMPLE

Triangle begins:

1, 0,

3, 0,

3, 8, 1,

25, 28, 7,

17, 155, 143, 45,

259, 1005, 933, 323,

131, 2770, 7488, 7150, 2621,

3177, 27978, 64164, 62310, 23811,

1281, 51433, 294602, 619986, 607445, 239653,

...