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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

%I A110952 #20 Apr 03 2025 11:11:02
%S A110952 1,3,3,6,11,6,10,26,26,10,15,50,71,50,15,21,85,155,155,85,21,28,133,
%T A110952 295,379,295,133,28,36,196,511,799,799,511,196,36,45,276,826,1519,
%U A110952 1849,1519,826,276,45,55,375,1266,2674,3829,3829,2674,1266,375,55,66,495,1860
%N 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.
%C A110952 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.
%H A110952 Joseph Adams, <a href="/A110952/b110952.txt">Table of n, a(n) for n = 3..4755</a>
%F A110952 T(n,k) = k*C(n,k+1) - C(n,k) + 1.
%e A110952 Triangle (beginning with n=3, k=1) is:
%e A110952    1
%e A110952    3  3
%e A110952    6 11  6
%e A110952   10 26 26 10
%e A110952   15 50 71 50 15
%e A110952   ...
%e A110952 For 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 A110952 Cf. A045618, A000217, A051925, A001701, A112858.
%K A110952 easy,nonn,tabl
%O A110952 3,2
%A A110952 _David Scambler_, Nov 22 2006