The number of 1...davoiding permutations of length d+r for SYMBOLIC d but numeric r
By
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger
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First Written: April 9, 2015.
Last update (of this webpage): June 23, 2015
To IRA GESSEL, with Admiration and Friendship, on Turning MillionYearsOld (in Base 2)
We use the RobinsonSchensted correspondence, followed by symbolcrunching, in order to derive explicit expressions for
the quantities mentioned in the title. We follow it by number crunching, in order to compute the
first terms of these sequences. As an encore, we cleverly implement Ira Gessel's celebrated determinant formula
for the generating functions of these sequences, to crank out many terms.
This modest tribute is dedicated to one of the greatest enumerators alive today (and definitely the most modest one!),
Ira Martin Gessel, who is turning 64 yearsold today. Ira, may you continue to produce beautiful (and useful)
combinatorics, at least 64 more years!
Maple Package
Some Input and Output files for the Maple package Gessel64

If you want to see the explicit expressions for G_{d}(r+d), the number of
permutations of d+r avoiding an increasing sequence of length d, valide for d larger than r1, for r from 1 to 30,
the input yields
the output

If you want to see the first 2d+1 terms of the sequences G_{d}(n), for d from 3 to 30,
the input yields
the output

If you want to see the first 100 terms of the sequences G_{d}(n), for d from 3 to 20,
the input yields
the output

[Added June 23, 2015]
If you want to see the first 9 leading terms in the polynomial expression (of degree 2k in m) for the number of permutations of
length m+k containing at least one increasing subsequence of length m
the input yields
the output
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page