# The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r

By
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger

.pdf   .ps   .tex

First Written: April 9, 2015.

Last update (of this webpage): June 23, 2015

To IRA GESSEL, with Admiration and Friendship, on Turning Million-Years-Old (in Base 2)
We use the Robinson-Schensted correspondence, followed by symbol-crunching, 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 years-old today. Ira, may you continue to produce beautiful (and useful) combinatorics, at least 64 more years!

# Maple Package

• Gessel64: a Maple package to implement the algorithms of our paper.
[Last Update: June 23, 2015].

# Some Input and Output files for the Maple package Gessel64

• If you want to see the explicit expressions for Gd(r+d), the number of permutations of d+r avoiding an increasing sequence of length d, valide for d larger than r-1, for r from 1 to 30,

the input   yields the output

• If you want to see the first 2d+1 terms of the sequences Gd(n), for d from 3 to 30,

the input   yields the output

• If you want to see the first 100 terms of the sequences Gd(n), for d from 3 to 20,

the input   yields the output