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

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 editors-in-chief, 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 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 co-EIC Akihiro MUNEMASA and Steering Committee Head Sara BILLEY of the Electronic Journal ALCO (Algebraic Combinatorics) Seriously Erred in Rejecting (w/o Refereeing) the Kauers-Zeilberger 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! ... (k-1)!/((n+k-1)!(n+k-2)!... 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 "Goulden-Jackson" 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

Sample Input and Output Files for YoungT.txt

Sample Input and Output Files for Tableaux3R.txt

Papers of Doron Zeilberger

Doron Zeilberger's Home Page