A145889 Number of even entries that are followed by a smaller entry in all permutations of {1,2,...,n}.

0, 1, 2, 24, 96, 1080, 6480, 80640, 645120, 9072000, 90720000, 1437004800, 17244057600, 305124019200, 4271736268800, 83691159552000, 1339058552832000, 28810681675776000, 518592270163968000, 12164510040883200000

Offset: 1

Author

Emeric Deutsch, Nov 16 2008

Keywords

• nonn
• easy

Comments

a(n) = Sum(k*A134434(n,k), k=0..floor(n/2)).

The average of the number of even entries that start a descent over all permutations of {1,2,...n} is (1/n)[floor(n/2)]^2.

Examples

			a(3)=2 because the permutations of {1,2,3} are 123, 132, 2'13, 231, 312 and 32'1 with the even entries that start a descent marked.
		

References

• S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.

Crossrefs

Cf. A134434, A145890.

Programs

Maple

a:=proc(n) if `mod`(n,2)=0 then (1/4)*n*factorial(n) else (1/4)*(n-1)^2*factorial(n-1) end if end proc: seq(a(n),n=1..20);

Formula

a(2n) = n(2n)!/2; a(2n+1) = n^2*(2n)!.

D-finite with recurrence (-4*n+11)*a(n) +(9*n-25)*a(n-1) +(n-2)*(4*n^2-3*n-3)*a(n-2) -(n-2)*(n-3)*(5*n-7)*a(n-3)=0. - R. J. Mathar, Jul 31 2022