A152667 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of even entries (n >= 2, 1 <= k <= floor(n/2)). For example, the permutation 321756498 has 3 runs of even entries: 2, 64 and 8.

2, 6, 12, 12, 48, 72, 144, 432, 144, 720, 2880, 1440, 2880, 17280, 17280, 2880, 17280, 129600, 172800, 43200, 86400, 864000, 1728000, 864000, 86400, 604800, 7257600, 18144000, 12096000, 1814400, 3628800, 54432000, 181440000, 181440000, 54432000, 3628800

Offset: 2

Author

Emeric Deutsch, Dec 14 2008

Keywords

• nonn
• tabf

Comments

Sum of entries in row n is n! (=A000142(n)). Row n contains floor(n/2) entries.

T(n,1) = A152873(n).

Sum_{k>=1} k*T(n,k) = A152668(n).

Mirror image of A145892.

Examples

			T(4,2) = 12 because we have 1234, 3214, 1432, 3412, 2134, 2314 and their reverses.
Triangle starts:
    2;
    6;
   12,   12;
   48,   72;
  144,  432,  144;
  720, 2880, 1440;
		

Crossrefs

Cf. A000142, A152666, A152668, A145892, A152873.

Programs

Maple

ae := proc (n, k) options operator, arrow: factorial(n)^2*binomial(n+1, k)*binomial(n-1, k-1) end proc: ao := proc (n, k) options operator, arrow: factorial(n)*factorial(n+1)*binomial(n-1, k-1)*binomial(n+2, k) end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc; for n to 12 do seq(T(n, k), k = 1 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := If[EvenQ[n], ((n/2)!)^2*Binomial[n/2+1, k]*Binomial[n/2-1, k-1], ((n-1)/2)!*((n-1)/2+1)!*Binomial[(n-1)/2-1, k-1]*Binomial[(n-1)/2+2, k]];
Table[T[n, k], {n, 2, 12}, {k, 1, Floor[n/2]}] // Flatten (* Jean-François Alcover, Sep 24 2024 *)

Formula

T(2n,k) = (n!)^2 * binomial(n+1,k) binomial(n-1,k-1);

T(2n+1,k) = n!*(n+1)!*binomial(n-1,k-1)*binomial(n+2,k) (n >= 1).