A177264 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the first entry in the last block (1<=k<=n).

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

Offset

1,2

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 A177263.

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.

Formula

T(n, k) = (n-1)! - (k-2)! if 2 <= k <= n, otherwise T(n, 1) = 0! + 1! + ... + (n-1)! = A003422(n).

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

T(n, k) = A177263(n, n-k+1) (mirror image).

Example

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

Maple

T := proc (n, k) if 2 <= k and k <= n then factorial(n-1)-factorial(k-2) elif k = 1 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==1, A003422[n], (n-1)! -(k-2)!];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, May 19 2024 *)

Prog

(Magma)
A003422:= func< n | (&+[Factorial(j): j in [0..n-1]]) >;
A177264:= func< n, k | k eq 1 select A003422(n) else Factorial(n-1) - Factorial(k-2) >;
[A177264(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 19 2024
(SageMath)
def A003422(n): return sum(factorial(j) for j in range(n))
def A177264(n, k): return A003422(n) if k==1 else factorial(n-1) - factorial(k-2)
flatten([[A177264(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, May 19 2024

Crossrefs

Cf. A000142, A003422, A177263.

Sequence in context: A264379 A090888 A154794 * A326758 A318366 A300252

Adjacent sequences: A177261 A177262 A177263 * A177265 A177266 A177267

Keyword

nonn,tabl

Author

Emeric Deutsch, May 16 2010

Status

approved