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). 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.

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

Offset

1,2

Comments

Sum of entries in row n is n!.

Mirror image of A177263.

T(n,1)=A003422(n).

References

A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.

Links

Table of n, a(n) for n=1..49.

Formula

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

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;

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

Crossrefs

Cf. 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