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.

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, 64, 0, 297, 345, 283, 195, 119, 63, 128, 0, 1001, 1166, 965, 676, 425, 246, 127, 256, 0, 3432, 4004, 3333, 2359, 1506, 894, 501, 255, 512

Offset: 1

Author

David Callan, May 31 2016

Keywords

• nonn
• tabl

Comments

It appears that each column, other than the first, has asymptotic growth rate of 4.

Examples

			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).
Triangle begins:
1
0   2
0   1   4
0   3   3   8
0   9  10   7  16
0, 28, 32, 25, 15, 32
...
		

Crossrefs

Except for the initial term, column 2 is A000245, column 3 is A071718, and row sums are A000108.

Programs

Mathematica

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]]]]
u[1, 1] = 1; u[2, 2] = 2;
u[n_, 1] /; n > 1 := 0; u[n_, k_] /; n < 1 || k < 1 || k > n := 0;
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];
Table[u[n, k], {n, 10}, {k, n}]

Formula

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).