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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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

# 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.