Computing Determinants Involving Stirling Numbers

By Tewodros Amdeberhan and Shalosh B. Ekhad

.pdf    .tex

Written: June 17, 2022.

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and Tewodros Amdeberhan's web-site.

We use the Jacobi-Trudi formula for Schur polynomials, and Maple, to explicitly compute explicit expressions for many determinants involving Stirling numbers of both kinds.

# Maple package

• StirlingDet.txt, To compute many determinants involving Stirling numbers.

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 (a-b)n . SSYTs form what I call a 'column-closed 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 .

Articles of Doron Zeilberger