Riffle shuffle permutation

In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of $n$ items that can be obtained by a single riffle shuffle, in which a sorted deck of $n$ cards is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). Beginning with an ordered set (1 rising sequence), mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences.^[1] The permutations with 1 rising sequence are the identity permutations.

As a special case of this, a $(p,q)$ -shuffle, for numbers $p$ and $q$ with $p+q=n$ , is a riffle in which the first packet has $p$ cards and the second packet has $q$ cards.^[2]

^ Aldous, David; Diaconis, Persi (1986), "Shuffling cards and stopping times" (PDF), The American Mathematical Monthly, 93 (5): 333–348, doi:10.2307/2323590, JSTOR 2323590, MR 0841111
^ Weibel, Charles (1994). An Introduction to Homological Algebra, p. 181. Cambridge University Press, Cambridge.

[1] Aldous, David; Diaconis, Persi (1986), "Shuffling cards and stopping times" (PDF), The American Mathematical Monthly, 93 (5): 333–348, doi:10.2307/2323590, JSTOR 2323590, MR 0841111

[Weibel-2] Weibel, Charles (1994). An Introduction to Homological Algebra, p. 181. Cambridge University Press, Cambridge.

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