A145892 -id:A145892 - cp's OEIS frontend

A145891 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k adjacent pairs of the form (odd,even) (0<=k<=floor(n/2)).

1, 1, 1, 1, 2, 4, 4, 16, 4, 12, 72, 36, 36, 324, 324, 36, 144, 1728, 2592, 576, 576, 9216, 20736, 9216, 576, 2880, 57600, 172800, 115200, 14400, 14400, 360000, 1440000, 1440000, 360000, 14400, 86400, 2592000, 12960000, 17280000, 6480000, 518400

Offset: 0

Emeric Deutsch, Nov 30 2008

Also number of permutations of {1,2,...,n} having k adjacent pairs of the form (even,odd). Example: T(3,1) = 4 because we have 123, 213, 231 and 321.
Row n contains 1+floor(n/2) entries.
Mirror image of A134434.
Sum of entries in row n = n! = A000142(n).
Sum_{k>=0} k*T(n,k) = A077613(n).

			T(3,1) = 4 because we have 123, 132, 312 and 321.
Triangle starts:
   1;
   1;
   1,   1;
   2,   4;
   4,  16,   4;
  12,  72,  36;
  36, 324, 324, 36;
  ...

Cf. A000142, A077613, A134434, A134435, A145892.

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

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

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

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.

Original entry on oeis.org

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

Emeric Deutsch, Dec 14 2008

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.

			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;

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

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

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 *)

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).

cp's OEIS Frontend

A145891 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k adjacent pairs of the form (odd,even) (0<=k<=floor(n/2)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula