Determinantal formula for generalized riffle shuffle

Nakano, Fumihiko; Sadahiro, Taizo

doi:10.1016/j.disc.2021.112599

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Eigenvalues and Their Multiplicities1

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2022

2023

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Cited by 3 publications

(2 citation statements)

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“…However, k a p is different from the Stirling-Frobenius cycle number of parameter p which appears in the analysis of a p-version of the riffle shuffle [9,10].…”

Section: Eigenvalues and Their Multiplicitiesmentioning

confidence: 91%

See 1 more Smart Citation

Top to random shuffles on colored permutations

Nakano¹,

Sadahiro²,

Sakurai³

2022

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A deck of n cards are shuffled by repeatedly taking off the top card, flipping it with probability 1/2, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group B n of signed permutations. We show that the eigenvalues of the transition probability matrix are 0, 1/n, 2/n, . . . , (n − 1)/n, 1 and the multiplicity of the eigenvalue i/n is equal to the number of the signed permutation having exactly i fixed points. We show the similar results also for the colored permutations. Further, we show that the mixing time of this Markov chain is n log n, same as the ordinary 'top-to-random' shuffles without flipping the cards. The cut-off is also analyzed by using the asymptotic behavior of the Stirling numbers of the second kind.

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“…However, k a p is different from the Stirling-Frobenius cycle number of parameter p which appears in the analysis of a p-version of the riffle shuffle [9,10].…”

Section: Eigenvalues and Their Multiplicitiesmentioning

confidence: 91%

“…(3) For a generalized riffle shuffle (i.e., the riffle shuffle on G n,p ), the multiplicity of eigenvalues are equal to the Stirling-Frobenius cycle number [10].…”

Section: Remarkmentioning

confidence: 99%