# The order of a perfect k-shuffle on a moded-out deck

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## Abstract

When a deck of n cards is shuffled so that the shuffle is some fixed permutation on n objects, i.e., an element of 5, it is natural to ask how many times this shuffle must be performed before the deck will return to its original configuration. The number of shuffles needed for this to happen is called the order of the shuffle. A perfect 2-shuffle involves dividing an even deck in two halves and then interlacing the cards so they alternate, i.e., how a two- armed person would perfectly shuffle the deck. A precise definition of how this concept would be extended to a A:-armed person is given. For a deck of size ks, it is known that the order of the simplest perfect A:-shuffle is the order of k modulo ks — 1, i.e., the minimum positive integer d such that A:'^ = 1 (mod ks — 1).

This dissertation deals with decks in which the cards are not all distinct. The definition of the order of a perfect A;-shuffle is extended to "moded-out" decks, i.e., decks in which the cards repeat in blocks of equal length. This dissertation gives a theorem which gives the order of a perfect A;-shuffle on a "moded-out" deck of size ks. Also given are consequences of the theorem that are related to what is known about the order of a perfect A;-shuffle on a regular deck.