%I #7 Mar 31 2012 20:25:47
%S 1,3,3,6,11,6,10,26,26,10,15,50,71,50,15,21,85,155,155,85,21,28,133,
%T 295,379,295,133,28,36,196,511,799,799,511,196,36,45,276,826,1519,
%U 1849,1519,826,276,45,55,375,1266,2674,3829,3829,2674,1266,375,55,66,495,1860
%N 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.
%C 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.
%F T(n,k) = k*C(n,k+1) - C(n,k) + 1
%e Triangle (beginning with n=3, k=1) is:
%e 1
%e 3 3
%e 6 11 6
%e 10 26 26 10
%e 15 50 71 50 15
%e 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}
%Y Cf. A045618, A000217, A051925, A001701, A112858.
%K easy,nonn,tabl
%O 3,2
%A _David Scambler_, Nov 22 2006