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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
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 (list; table; graph; refs; listen; history; text; internal format)
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: A261251 A039991 A081265 * A108643 A133838 A182138

Adjacent sequences:  A273818 A273819 A273820 * A273822 A273823 A273824

KEYWORD

nonn,tabl

AUTHOR

David Callan, May 31 2016

STATUS

approved

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Last modified December 5 20:45 EST 2019. Contains 329779 sequences. (Running on oeis4.)