A177263 - OEIS

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

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OFFSET

1,3

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.

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)! - (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).

EXAMPLE

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:

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;

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 *)

PROG

(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

(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)