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). 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, 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

Offset

1,3

Comments

Sum of entries in row n is n!.

Mirror image of A177264.

T(n,1)=A094304(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)!-(n-k-1)! if k<=n-1; T(n,n)=0!+1!+...+(n-1)!.

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:
1;
0,2;
1,1,4;
4,5,5,10;
18,22,23,23,34;

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

Crossrefs

Cf. A094304, A177264.

Sequence in context: A144460 A336996 A222541 * A357050 A197380 A057785

Adjacent sequences: A177260 A177261 A177262 * A177264 A177265 A177266

Keyword

nonn,tabl

Author

Emeric Deutsch, May 16 2010

Status

approved