Counting Standard Young Tableaux With Restricted Runs
By Manuel Kauers and Doron Zeilberger
.pdf
.ps
.tex
First Written: June 21, 2020. This version (version 2): Aug. 8, 2020.
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)
Dedicated to the legendary enumerators Ian Goulden and David Jackson
Added July 27, 2020:
This article was originally submitted to the electronic journal ``Algebraic Combinatorics''
founded by Goulden and Jackson, following a solicitation for a special issue
in honor of Goulden and Jackson. On July 27, 2020 we got an email message from one of the
editorsinchief, Akihiro Munemasa, informing us that, after an initial review, it is ``unlikely
to meet the standards of depth and originality that the journal is seeking''. Consequently this
article will remain in the `Personal journal of Shalosh B. Ekhad and Doron Zeilberger', the homepage of Manuel Kauers, and
of course arxiv.org. Let the readers decide about its depth and originality.
We are disappointed at the editors of Algebraic Combinatorics for their erroneous and misguided decision.
Regarding this unfortunate decision, read this recent opinion .
Added Aug. 11, 2018: read the appendix to the above opinion
Why coEIC Akihiro MUNEMASA and Steering Committee Head Sara BILLEY of the Electronic Journal ALCO (Algebraic Combinatorics) Seriously Erred in Rejecting (w/o Refereeing) the KauersZeilberger submission about Counting Young Tableaux".
Added Oct. 22, 2020: Watch my lecture at the Manuel Kauers seminar, Oct. 22, 2020,
How the Beautiful Duckling of Enumerative Combinatorics turned into the Ugly Swan of Algebraic Combinatorics
The number of Young Tableaux whose shape is a k by n rectangle is
famously (nk)! 0! ... (k1)!/((n+k1)!(n+k2)!... n!) implying
that for each specific k, that sequence satisfies a linear recurrence equation with polynomial
coefficients of the FIRST order. But what about counting Young tableaux where
certain "run lengths" are forbidden? Then things seem to get much more complicated.
In this tribute to the legendary enumerative pair "GouldenJackson" we
investigate these intriguing sequences. We conclude with four intriguing conjectures, and
will be happy to donate $200 dollars to the OEIS in honor of the first prover,
for each of them.
Maple packages
C programs

Gseq.c
A C program to compute many terms of the Gsequence of our paper modulo primes

Hseq.c
A C program to compute many terms of the Hsequence of our paper modulo primes
Sample Input and Output Files for YoungT.txt

If you want to see various such sequences in 2, 3, and 4 dimensions
the input file
yields the
output file

If you want to see the first 100 terms of sequences of various sequences enumerating 3 by n Young tableaux
where each row is either unrestricted or can't have a run of length 1 (altogether 8 cases), together with
their ESTIMATED (nonrigorous) asympototics
the input file
yields the
output file

If you want to see the first 100 terms of sequences of various sequences enumerating 3 by n Young tableaux
where each row is not allowed to have runlengths that belong to some arithmetical progression
and their ESTIMATED (nonrigorous) asympototics
the input file
yields the
output file

Added Aug. 10, 2020: As beautifully proved in the paper, the analogous family of sequences
counting Lattice Walks from the origin to [n,...,n] with restricted runs is
Precursive, albeit possibly fairly complex.
To see the linear recurrence of order 13 (and degree 10) satisfied by the
sequence: number of lattice walks from [0,0,0] to [n,n,n] where each run (consecutive segment in one direction)
is of length 2.
the input file
yields the
output file
Sample Input and Output Files for Tableaux3R.txt
Papers of Doron Zeilberger
Doron Zeilberger's Home Page