There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]

TL;DR

This paper demonstrates how the Even-Gillis integral formula, combined with Wilf-Zeilberger algorithms, can efficiently compute the number of derangements of a multiset, including extremely large counts, extending to other complex combinatorial problems.

Contribution

It introduces a novel computational approach that leverages special functions and algorithmic proof theory to count derangements and related combinatorial quantities efficiently.

Findings

Successfully computed extremely large derangement counts.

Extended the method to other complex combinatorial problems.

Showed the effectiveness of combining integral formulas with algorithmic proof techniques.

Abstract

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.