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A273821 Triangle read by rows: T(n,k) is the number of 123-avoiding 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. 0

%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 123-avoiding 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 123-avoiding 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

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Last modified May 17 04:58 EDT 2021. Contains 343964 sequences. (Running on oeis4.)