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A134433 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1 <= k <= n).

%I A134433 #14 Nov 15 2019 23:08:44
%S A134433 1,0,2,0,1,5,0,2,6,16,0,6,16,33,65,0,24,60,114,196,326,0,120,288,522,
%T A134433 848,1305,1957,0,720,1680,2952,4632,6850,9786,13700,0,5040,11520,
%U A134433 19800,30336,43710,60672,82201,109601
%N A134433 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1 <= k <= n).
%C A134433 T(n,n) = A000522(n-1) (number of arrangements of {1,2,...,n-1}).
%C A134433 T(n,2) = (n-2)! for n >= 3.
%C A134433 Sum_{k=1..n} k*T(n,k) = A056542(n+1).
%H A134433 Michael De Vlieger, <a href="/A134433/b134433.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened.)
%F A134433 T(n,k) = Sum_{j=1..k-1} (n-j-1)!*(k-j)*binomial(k-1,j-1) for k < n;
%F A134433 T(n,n) = (n-1)!*Sum_{j=0..n-1} 1/j!.
%e A134433 T(4,3)=6 because we have 3124, 3142, 3214, 3241, 1324 and 2314.
%e A134433 Triangle starts:
%e A134433   1;
%e A134433   0,   2;
%e A134433   0,   1,   5;
%e A134433   0,   2,   6,  16;
%e A134433   0,   6,  16,  33,  65;
%e A134433   0,  24,  60, 114, 196, 326;
%p A134433 T:=proc(n,k): if k < n then sum(factorial(k-1)*factorial(n-j-1)/(factorial(j-1)*factorial(k-j-1)), j=1..k-1) elif k = n then factorial(n-1)*(sum(1/factorial(j), j = 0 .. n-1)) else 0 end if end proc: for n to 9 do seq(T(n, k),k=1..n) end do; # yields sequence in triangular form
%t A134433 Table[If[k < n, Sum[(n - j - 1)!*(k - j)*Binomial[k - 1, j - 1], {j, k - 1}], (n - 1)!*Sum[1/j!, {j, 0, n - 1}]], {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Nov 15 2019 *)
%Y A134433 Cf. A000522, A056542.
%K A134433 nonn,tabl
%O A134433 1,3
%A A134433 _Emeric Deutsch_, Nov 22 2007