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

1,3

Comments

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

Links

Table of n, a(n) for n=1..55.

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).

Example

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
...

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}]

Crossrefs

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

Sequence in context: A342909 A081265 A039991 * A363519 A108643 A133838

Adjacent sequences: A273818 A273819 A273820 * A273822 A273823 A273824

Keyword

nonn,tabl

Author

David Callan, May 31 2016

Status

approved