%I
%S 1,0,2,0,1,4,0,3,3,8,0,9,10,7,16,0,28,32,25,15,32,0,90,104,84,56,31,
%T 64,0,297,345,283,195,119,63,128,0,1001,1166,965,676,425,246,127,256,
%U 0,3432,4004,3333,2359,1506,894,501,255,512
%N Triangle read by rows: T(n,k) is the number of 123avoiding permutations p of [n] (A000108) such that k is maximal with the property that the k largest entries of p, taken in order, avoid 132.
%C It appears that each column, other than the first, has asymptotic growth rate of 4.
%F G.f.: Sum_{n>=1, 1<=k<=n} T(n,k) x^n y^k = C(x)  1 + ((1  y) (1  x y) (1  (1  x y)C(x)))/((1  2 x y) (1  y + x y^2) ) where C(x) = 1 + x + 2x^2 + 5x^3 + ... is the g.f. for the Catalan numbers A000108 (conjectured).
%e For example, for the 123avoiding permutation p = 42513, the 3 largest entries, 453, avoid 132 but the 4 largest entries, 4253, do not, and so p is counted by T(5,3).
%e Triangle begins:
%e 1
%e 0 2
%e 0 1 4
%e 0 3 3 8
%e 0 9 10 7 16
%e 0, 28, 32, 25, 15, 32
%e ...
%t Map[Rest, Rest[Map[CoefficientList[#, y] &, CoefficientList[ Normal[Series[ c  1 + ((1  y) (1  x y) (1  (1  x y) c ))/((1  2 x y) (1  y + x y^2)) /. {c :> (1  Sqrt[1  4 x])/(2 x)}, {x, 0, 10}, {y, 0, 10}]], x]]]]
%t u[1, 1] = 1; u[2, 2] = 2;
%t u[n_, 1] /; n > 1 := 0; u[n_, k_] /; n < 1  k < 1  k > n := 0;
%t u[n_, k_] /; n >= 3 && 2 <= k <= n := u[n, k] = 3 u[n  1, k  1]  2 u[n  2, k  2] + u[n, k + 1]  2 u[n  1, k] + If[k == 2, CatalanNumber[n  2], 0];
%t Table[u[n, k], {n, 10}, {k, n}]
%Y Except for the initial term, column 2 is A000245, column 3 is A071718, and row sums are A000108.
%K nonn,tabl
%O 1,3
%A _David Callan_, May 31 2016
