Refined Asymptotics and Explicit Recurrences for the numbers of Young tableaux in the (k,l) hook for k+l ≤ 5
By Shalosh B. EKHAD and Amitai REGEV
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.tex
Exclusively published in the Personal Journal of Ekhad and Zeilberger and arxiv.org .
Written: July 24, 2010.
This is an etude in Experimental semi-rigorous (rigorizable!)
mathematics. The leading asymptotics was brilliantly derived
by Allan Berele and Amitai Regev for general hooks H(k,l) and general powers z,
but what about more refined asymptotics?
For small k and l, one can "guess" a linear recurrence (since we live in the holonomic ansatz) and
using the Birkhoff-Trjitzinsky method, beautifully implemented in
Doron Zeilberger's Maple packageAsyRec
(that has been incorporated into the present Maple package), we computed
amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev asymptotic formula, and especially
the impressive constant in front!
Maple Package
VERY IMPORTANT: This article is accompanied by Maple
package
Input and Output for HOOKER
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(2,1), the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(2,1), the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(2,2), the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(2,2), the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(3,0),
(in other words SYT with at most three rows, famously proved by Amitai Regev
to be equal to the Motzkin numbers)
the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(3,0),
(that thanks fo the Robinson-Schenstead correspondence equals the number of 1234-avoiding permutations)
the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(3,1), the
input
gives the output.
-
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(3,1), the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(3,2), the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(3,2), the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(4,0),
(By the Robinson-Schenstead correspondence,
this is also the number of 12345-avoiding involutions)
the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(4,0),
(By the Robinson-Schenstead correspondence,
this is also the number of 12345-avoiding permutations)
the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(4,1),
the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(4,1),
the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the Standard Young Tableaux contained in the hook H(5,0),
(By the Robinson-Schenstead correspondence,
this is also the number of 123456-avoiding involutions)
the
input
gives the output.
-
To see the first 200 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of the squares of f(λ)
over all λ contained in the hook H(5,0),
(By the Robinson-Schenstead correspondence,
this is also the number of 123456-avoiding permutations)
the
input
gives the output.
-
[Added Aug. 30, 2010]
To see the first 100 terms, the linear recurrence, and the
refined asymptotics to order 10 of the sequence
enumerating the sum of f(2 λ)
over all 2λ (twice λ) with shapes λ with
at most 3 rows,
confirming Amitai Regev's amazing asymptotic formula in his paper
Asymptotics of Young tableaux in the strip, the d-sums
(second version, correcting a typo in a previous
version that Shalosh B. Ekhad detected!)
the
input
gives the output.
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