A134435 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k odd entries that are followed by a smaller entry (n >= 0, k >= 0).

1, 1, 2, 2, 4, 12, 12, 12, 72, 36, 144, 432, 144, 144, 1728, 2592, 576, 2880, 17280, 17280, 2880, 2880, 57600, 172800, 115200, 14400, 86400, 864000, 1728000, 864000, 86400, 86400, 2592000, 12960000, 17280000, 6480000, 518400

Offset

0,3

Comments

Row n has ceiling(n/2) entries (for n>0). T(2n,0) = T(2n+1,0) = n!*(n+1)! = A010790(n).

T(n,k) is also the number of permutations of {1,2,...,n} having k adjacent pairs of the form (odd, odd) (0 <= k <= ceiling(n,2)-1). Example: T(3,1)=4 because we have 132, 213, 312 and 231. - Emeric Deutsch, Dec 14 2008

Links

Table of n, a(n) for n=0..36.

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

Formula

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

Example

T(3,1) = 4 because we have 132, 312, 231 and 321.
Triangle starts:
    1;
    1;
    2;
    2,   4;
   12,  12;
   12,  72,  36;
  144, 432, 144;
  ...

Maple

T:=proc(n, k) if `mod`(n, 2)=0 then binomial((1/2)*n-1, k)*binomial((1/2)* n+1, k+1)*factorial((1/2)*n)^2 elif `mod`(n, 2)=1 then factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial((1/2)*n-1/2, k)*binomial((1/2)* n+1/2, k) else 0 end if end proc: for n from 0 to 11 do seq(T(n, k), k=0..max(0, ceil((1/2)*n)-1)) end do; # yields sequence in triangular form

Mathematica

T[n_, k_]:=If[EvenQ[n], ((n/2)!)^2Binomial[n/2-1, k]Binomial[n/2+1, k+1], ((n-1)/2)!((n+1)/2)!Binomial[(n-1)/2, k]Binomial[(n+1)/2, k]]; Table[T[n, k], {n, 11}, {k, 0, Floor[(n-1)/2]}]//Flatten (* Stefano Spezia, Jul 12 2024 *)

Crossrefs

Bisection of column k=0 gives A010790.

Row sums give A000142.

Cf. A134434.

Sequence in context: A285944 A112473 A353131 * A136718 A362709 A112362

Adjacent sequences: A134432 A134433 A134434 * A134436 A134437 A134438

Keyword

nonn,tabf,changed

Author

Emeric Deutsch, Nov 22 2007

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Jul 12 2024

Status

approved