The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r occurrences of each of 1,2, ..., n are Always Algebraic
By
Nathaniel Shar and Doron Zeilberger
.pdf
.ps
.tex
[Appeared in Annals of Combinatorics 20 (2016), 387-396]
First Written: Nov. 18, 2014 ; This version: Nov. 26, 2014 [Thanks to Robin Chapman]
The set of permutations (alias words in the alphabet {1, ..., n} with exactly 1 occurrence
of each letter) that do not have an increasing subsequence of length 3
is famously enumerated by the ubiquitous
Catalan numbers,
whose generating function, C(x)
famously satisfies the algebraic equation
C(x)=1+xC(x)^{2} .
Recently, Bill Chen, together with his disciples
Alvin Dai and Robin Zhou, discovered, and very elegantly
proved, an
algebraic equation satisfied by the generating function enumerating 123-avoiding words with
two occurrences of each of 1, ..., n.
[Sequence A220097 of the bible]
Inspired by the Chen-Dai-Zhou result, we present an algorithm for finding such an algebraic equation for the
ordinary generating function enumerating 123-avoiding words with exactly r occurrences of
each of 1, ... n for any positive integer r, thereby proving that they are algebraic
and not merely D-finite (a fact that is promised by WZ theory).
Our algorithm consists of presenting an algebraic enumeration scheme, combined with the Buchberger algorithm.
Added Nov. 26, 2014: This new version states Robin Chapman's conjectured expression for the constant C_{r}
in front of the conjectured asymptotic expression for w_{r}(n) in the article. (Proved rigoroulsy for 1 ≤ r ≤ 5,
except for the constant, that officially is still conjectured even for r=2, but could be easily proved rigorously using
standard techniques described so nicely in the Flajolet-Sedgewick bible.]
Added Dec. 30, 2014. The $100 dollar conjecure was
generalized, and
this more general conjecture (that implies the conjecture made in the presend article) was
proved by Guillaume Chapuy. A donation to the OEIS was made.
Maple Package
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