By Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger
Posted: Jan. 25, 2021
[Published in Enumerative Combinatorics and Applidcations v. 1, issue 3, article S2A17]
In fond memoroy of Joe Gillis (3 Aug. 1911- 18 Nov. 1993), who taught us that Special Functions Count
In this memorial tribute to Joe Gillis, who taught us that Special Functions count, we show how the seminal Even-Gillis integral formula for the number of derangements of a multiset, in terms of Laguerre polynomials, can be used to efficiently compute not only the number of the title, but much harder ones, when it is interfaced with Wilf-Zeilberger algorithmic proof theory.
1(repeated k times), 2(repeated k times), ..., n (repeated k times)
for k between 1 (the classical Euler case) and k=19 (very complicated!)
as well the 2000-th term
The input file
yields the output file
1(repeated k times), 2(repeated k times), ..., n (repeated k times)
for n between 2 (trivially identically 1 (why?)) and n=9 (very complicated!)
as well the 2000-th term
The input file
yields the output file
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