A177263 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the last entry in the first block (1<=k<=n).

1, 0, 2, 1, 1, 4, 4, 5, 5, 10, 18, 22, 23, 23, 34, 96, 114, 118, 119, 119, 154, 600, 696, 714, 718, 719, 719, 874, 4320, 4920, 5016, 5034, 5038, 5039, 5039, 5914, 35280, 39600, 40200, 40296, 40314, 40318, 40319, 40319, 46234, 322560, 357840, 362160, 362760, 362856, 362874, 362878, 362879, 362879, 409114

Offset: 1

Author

Emeric Deutsch, May 16 2010

Keywords

• nonn
• tabl

Comments

A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 45123867 has 4 blocks: 45, 123, 8, and 67.

Mirror image of A177264.

Examples

			T(4,2)=5 because we have 12-4-3, 2-1-34, 2-1-4-3, 2-4-1-3, and 2-4-3-1 (the blocks are separated by dashes).
Triangle starts:
     1;
     0,    2;
     1,    1,    4;
     4,    5,    5,   10;
    18,   22,   23,   23,   34;
    96,  114,  118,  119,  119,  154;
   600,  696,  714,  718,  719,  719,  874;
  4320, 4920, 5016, 5034, 5038, 5039, 5039, 5914;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.

Crossrefs

Cf. A000142, A003422, A094304, A177264.

Programs

Magma

A003422:= func< n | (&+[Factorial(j): j in [0..n-1]]) >;
A177263:= func< n,k | k eq n select A003422(n) else Factorial(n-1) - Factorial(n-k-1) >;
[A177263(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 19 2024

Maple

T := proc (n, k) if k <= n-1 then factorial(n-1)-factorial(n-k-1) elif k = n then sum(factorial(j), j = 0 .. n-1) else 0 end if end proc: for n to 10 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

A003422[n_]:= Sum[j!, {j,0,n-1}];
T[n_, k_]:= If[k==n, A003422[n], (n-1)! -(n-k-1)!];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 19 2024 *)

SageMath

def A003422(n): return sum(factorial(j) for j in range(n))
def A177263(n,k): return A003422(n) if k==n else factorial(n-1) - factorial(n-k-1)
flatten([[A177263(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, May 19 2024

Formula

T(n, k) = (n-1)! - (n-k-1)! if k <= n-1, otherwise T(n, n) = 0!+1!+...+(n-1)! = A003422(n).

T(n, 1) = A094304(n).

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

T(n, k) = A177264(n, n-k+1) (mirror image).