Some Deep and Original Questions about the "critical exponents" of Generalized Ballot Sequences
By Shalosh B. Ekhad and Doron Zeilberger
.pdf
.tex
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)
Written: April 4, 2021.
We numerically estimate the critical exponents of certain enumeration sequences
that naturally generalize the famous Catalan and superCatalan sequences, and raise
deep and original questions about their exact values, and whether they
are rational numbers.
Added April 7, 2021:
Michael Wallner proved that indeed none of these
sequences are Precursive (in the 3D case) and none of the critical exponents are rational numbers,
completely answering (in the 3D case) the deep and original questions raised in the article.
read his insightful Email message
Michael Wallner plans to write it up as a short note for the arxiv. As soon as it is there, I will
make the promised donation to the OEIS in his honor.
Added May 27, 2021: Michael Wallner just posted his
beautiful artice. A donation to the OEIS, in his honor, has been made.
Maple package
Sample Input and Output files for Capone.txt

If you want to see the first few terms, and estimated asymptotics, and most important,
"estimated" critical exponents ( of course they are all 3/2 in the 2d cases) for many 2D ballot sequences
the input file yields
the output file

If you want to see the first few terms, and estimated asymptotics, and most important,
"estimated" critical exponents for many 3D ballot sequences
the input file yields
the output file

If you want to see the first few terms, and estimated asymptotics, and most important,
"estimated" critical exponents for a few 4D ballot sequences
the input file yields
the output file

If you want to see the first 400 terms of the sequence enumerating walks in the 3D cubic lattice
from [0,0,0] to [n,2n,2n] such that it always stays in the region 2x ≥y ≥ z
that gives the estimate of 3.73122 for the cricial exponent
the input file yields
the output file

If you want to see many terms of sequences enumerating, in the 2D square lattice nstep walks that say in A*x ≥ B*y,
for A ≤ B ≤ (with gcd(A,B)=1)
with positive unit steps suggesting that the critical exponent (to each subsesequence for n mod A+B)
the critical exponent is 0
the input file yields
the output file

If you want to see many terms of sequences enumerating, in the 3D square lattice nstep walks that say in A*x ≥ B*y ≥ C*z
for A ≤ B ≤ C (gcd(A,B,C)=1) between
with estimates for the cricial exponents
the input file yields
the output file
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page