By Alon Regev, Amitai Regev, and Doron Zeilberger
[With great assistance by Shalosh B. Ekhad]
The aim of this note is to indicate that a similar statement seems to hold for the character tables
of the symmetric groups Sn. Just as important, it is a case-study
in using a computer algebra system to prove deep identities, way beyond the ability of mere humans.
Added Oct. 27, 2016: Christine Bessenrodt discovered (and proved) a
lovely refinement
Written: July 13, 2015
[Appeared in J. of Difference Equations and its Applications 22(2016), no. 2, 272-279.
DOI:10.1080/10236198.2015.1081386
]
In the classic "Concrete Math", by Graham, Patashnik and Knuth they state:
"The numbers in Pascal's triangle satisfy,
practically speaking, infinitely many identities, so it is
not too surprising that we can find some surprising relationships by looking closely."
Added later: read the
sequel,
Maple Package
Important: This article is accompanied by Maple
package
Χλ(μ)
where λ ranges over all hook shapes with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 14, the
Χλ(μ)
where λ ranges over all shapes with at most TWO rows, with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 14, the
Χλ(μ)
where λ ranges over all shapes with at most THREE rows, with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 9, the
Χλ(μ)
where λ ranges over all shapes with at most THREE rows, with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 7, the
Χλ(μ)
where λ ranges over all shapes with at most FOUR rows, with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 7, the
Χλ(μ)
where λ ranges over all shapes with at most FIVE rows, with n cells, and where μ is a fixed partition, μ0 (whose smallest part is larger than 1) followed by n-|μ0| ones, for all such μ0, with |μ0| ≤ 5, the