A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0<k<n-1.

1, 3, 3, 6, 11, 6, 10, 26, 26, 10, 15, 50, 71, 50, 15, 21, 85, 155, 155, 85, 21, 28, 133, 295, 379, 295, 133, 28, 36, 196, 511, 799, 799, 511, 196, 36, 45, 276, 826, 1519, 1849, 1519, 826, 276, 45, 55, 375, 1266, 2674, 3829, 3829, 2674, 1266, 375, 55, 66, 495, 1860

Offset

3,2

Comments

Permutations of [n] with exactly 2 descents and the descents are adjacent. Adjusting for initial index: row sums are A045618; first diagonal is A000217, the triangular numbers; 2nd diagonal is A051925; and 3rd diagonal is A001701, generalized Stirling numbers.

Links

Table of n, a(n) for n=3..60.

Formula

T(n,k) = k*C(n,k+1) - C(n,k) + 1

Example

Triangle (beginning with n=3, k=1) is:
1
3 3
6 11 6
10 26 26 10
15 50 71 50 15
e.g. n=5, k = 2, T(5,2) = 11 = permutations of [5] with first run 2 long and last run 5-2-1 = 2 long, namely {14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312}

Crossrefs

Cf. A045618, A000217, A051925, A001701, A112858.

Sequence in context: A049926 A298954 A169944 * A339539 A025250 A326498

Adjacent sequences: A110949 A110950 A110951 * A110953 A110954 A110955

Keyword

easy,nonn,tabl

Author

David Scambler, Nov 22 2006

Status

approved