Posted: Jan. 7, 2025

Dedicated to Mourad E.-H.Ismail (b. April 27, 1944) on his 80th birthday, with friendship and admiration

We use the Almkvist-Zeilberger algorithm, combined with a weighted version of the Even-Gillis Laguerre integral due to Foata and Zeilberger, in order to efficiently compute weight enumerators of multiset derangements according to the number of cycles. The present paper is inspired by important previous work by Mourad Ismail and his collaborators, done in the late 1970s, but still useful after all these years.

Maple package

• Mourad.txt, a Maple package that does weighted counting of Multiset Derangements.

Sample Input and Output for Mourad.txt

• If you want to see the linear recurrences for the weight-enumerators, according to the number of cycles of π permutations of a set of the form

  A₁ Union ... Union A_n

  where the A_i are pairwise disjoint and each has k members, and for each member x of A_i , π(x) is NOT in A_i, for k from 1 to 10

  The input file yields the output file

• If you want to see the linear recurrences for the weight-enumerators, according to the number of cycles of π permutations of a set of the form

  A₁ Union ... Union A_n

  where the A_i are pairwise disjoint and each has k members, and for each member x of A_i , π(x) is NOT in A_i, for k from 1 to 7, and for each case, the first 30 terms, as well as the case n=100, followed by the average, variance, and scaled moments for the latter case

  The input file yields the output file

• If you want to see the weight-enumerator according to the number of cycles of derangements of a standard deck of cards

  The input file yields the output file

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