Generalizations of an Expansion Formula for Top to Random Shuffles

Tian, Roger

doi:10.1007/s00026-016-0332-y

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…Now we use the fact k = n log n − n • log(λ + o(1)) and the estimate (13) to apply the dominated convergence theorem on (12), yielding the conclusion.…”

Section: Asymptotics On Stirling Numbersmentioning

confidence: 95%

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Top to random shuffles on colored permutations

Nakano¹,

Sadahiro²,

Sakurai³

2022

Preprint

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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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“…Now we use the fact k = n log n − n • log(λ + o(1)) and the estimate (13) to apply the dominated convergence theorem on (12), yielding the conclusion.…”

Section: Asymptotics On Stirling Numbersmentioning

confidence: 95%

“…The following lemma follows from a theorem in [12] which treats more general cases. However we present its elementary proof.…”

Section: Eigenvalues and Their Multiplicitiesmentioning

confidence: 99%

Top to random shuffles on colored permutations

Nakano¹,

Sadahiro²,

Sakurai³

2022

Preprint

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“…Remark 3.5. In unpublished notes, Garsia [11] (see also Tian [29]), studies the top-to-random shuffling operator, which is adjoint or transpose to the random-to-top operator. There he sketches a proof that its minimal polynomial is…”

Section: 2mentioning

confidence: 99%

Invariant Theory for the free left-regular band and a q-analogue

Brauner¹,

Commins²,

Reiner³

2022

Preprint

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We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups:• the free left-regular band on n letters, acted on by the symmetric group, and • one of its q-analogues, acted on by the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and q-Stirling numbers.We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs.

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