A097971 Number of alternating runs in all permutations of [n] (the permutation 732569148 has four alternating runs: 732, 2569, 91 and 148).

2, 10, 56, 360, 2640, 21840, 201600, 2056320, 22982400, 279417600, 3672345600, 51891840000, 784604620800, 12640852224000, 216202162176000, 3912561709056000, 74694359900160000, 1500289571708928000, 31627726106296320000, 698242876346695680000

Offset: 2

Author

Emeric Deutsch and Ira M. Gessel, Sep 07 2004

Keywords

• nonn

Comments

a(n) is also equal to the sum over all permutations p in S(n) of the number of elements in the set {(i, j): 0 < i < j < n+1 and |i - j| = |p(i) - p(j)|}.

Examples

			a(3) = 10 because the permutations 123, 132, 312, 213, 231, 321 have the following alternating runs: 123, 13, 32, 31, 12, 21, 13, 23, 31 and 321.
		

References

• M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 24-30.
• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1973, Vol. 3, pp. 46 and 587-8.

Crossrefs

Cf. A006157.

Programs

Maple

seq(n!*(2*n-1)/3, n=2..20);

Formula

a(n) = n!(2n-1)/3. E.g.f.: x^2*(3-x)/[3(1-x)^2]. a(n) = 2*A006157(n).