A177262 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} starting with exactly k consecutive integers (1<=k<=n).

1, 1, 1, 4, 1, 1, 18, 4, 1, 1, 96, 18, 4, 1, 1, 600, 96, 18, 4, 1, 1, 4320, 600, 96, 18, 4, 1, 1, 35280, 4320, 600, 96, 18, 4, 1, 1, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 36288000, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1

Offset: 1

Author

Emeric Deutsch, May 15 2010

Keywords

• nonn
• tabl

Examples

			T(4,2)=4 because we have 1243, 2314, 3412, and 3421.
Triangle starts:
      1;
      1,    1;
      4,    1,   1;
     18,    4,   1,  1;
     96,   18,   4,  1,  1;
    600,   96,  18,  4,  1,  1;
   4320,  600,  96, 18,  4,  1,  1;
  35280, 4320, 600, 96, 18,  4,  1,  1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000142 (row sums), A005165, A007489, A094258.

Programs

Magma

A177262:= func< n,k | k eq n select 1 else (n-k)*Factorial(n-k) >;
[A177262(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 18 2024

Maple

T := proc (n, k) if k = n then 1 elif k < n then factorial(n-k)*(n-k) else 0 end if end proc: for n to 11 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

T[n_, k_]:= (n-k+1)! -(n-k)! +Boole[k==n];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 18 2024 *)

SageMath

def A177262(n,k): return (n-k)*factorial(n-k) + int(k==n)
flatten([[A177262(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, May 18 2024

Formula

T(n, k) = (n-k)*(n-k)! if k < n, otherwise T(n,n) = 1.

T(n, 1) = A094258(n) = (n-1)!(n-1).

Sum_{k=1..n} T(n, k) = A000142(n) (row sums).

Sum_{k=1..n} k*T(n,k) = Sum_{j=1..n} j! = A007489(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A005165(n). - G. C. Greubel, May 18 2024