Computing Determinants Involving Stirling Numbers
By Tewodros Amdeberhan and Shalosh B. Ekhad
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.tex
Written: June 17, 2022.
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and Tewodros Amdeberhan's website.
We use the JacobiTrudi formula for Schur polynomials, and Maple, to explicitly compute explicit expressions for
many determinants involving Stirling numbers of both kinds.
Maple package
Added June 27, 2022: Per Alexandersson made the following interesting remark:
"I think your result is closely related to Lemma 3, in this
article,
when applied to Schur polynomials) of shape (ab)^{n} .
SSYTs form what I call a 'columnclosed family of fillings', so counting these under stretching of the shape will give a
sequence satisfying a linear recurrence. Your result, explicitly giving the denominator, is very similar to computing the characteristic polynomial of the recurrence.
Of course, since you specialize the variables, it might be that your numerator is smaller than the one predicted by Lemma 3 above; it would be interesting to actually see if this is the case.
Sample Input and Output

If you want to see a list of explicit expressions, in terms of n, of the determinant Stirling1(a+i,b+j), 1 ≤ i ≤j ≤ n for 1 ≤ b ≤ a ≤ 10,
as well as the (ordinary) generating functions in the variable q
the input file yields the
output file .
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