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.

Maple package

• MultiDer.txt, a Maple package that uses the Even-Gillis amazing formula for counting multi-set derangements in terms of Laguerre polynomials, that combined with symbolic computation can do so much!

Sample Input and Output for MultiDer.txt

• If you want to see the first 30 terms, and INHOMOGENEOUS linear recurrences satisfied by the sequences {F[n](k),n=1,2,3,...} (the number of derangements of the multi-set

  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

• If you want to see the first 30 terms, and INHOMOGENEOUS linear recurrences satisfied by the sequences {F[n](k),k=1,2,3,...} (the number of derangements of the multi-set

  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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