Top to random and reverse: analysis of a new descent algebra shuffle

Darij Grinberg; Jonathan Parlett

Top to random and reverse: analysis of a new descent algebra shuffle

T0 review · 0 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that composing any descent-algebra basis element with the reversing permutation yields a diagonalizable operator whose minimal polynomial is the product of distinct linear factors indexed by signed knapsack numbers.

desk verdict Solid new spectral theorem for w0-composed descent algebra elements, with a clean proof and a nice application to a reverse top-to-random shuffle. read the letter →

arxiv 2508.06740 v1 pith:6265EULQ submitted 2025-08-08 math.CO

Darij Grinberg , Jonathan Parlett This is my paper

classification math.CO MSC 05E1820C3060B1560J10

keywords descentalgebratop-to-randomshufflesignedknapsacknumberssetcompositionsfaceminimalpolynomialdiagonalizabilitysymmetricgroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies card shuffles obtained by composing the classical top-to-random shuffle with the order-reversing permutation $w_0$ of the deck, and more generally by composing any element $B_\alpha$ of the descent algebra with $w_0$. It proves that, over any field of characteristic zero, every such composition is diagonalizable, and that its minimal polynomial is the product of distinct linear factors $(x-k)$ where $k$ runs over the 'signed knapsack numbers' of set compositions. For the reverse top-to-random shuffle itself, the eigenvalues are explicit: all integers from $-n+4$ to $n-3$, together with $-n+2$, $0$, and $n$. This gives a complete spectral description of a natural shuffle, extending the well-understood behaviour of the ordinary top-to-random shuffle to its mirror version. The proof works by transporting the shuffle into the face algebra of the braid arrangement, where a length-filtration triangularization exhibits the eigenvalues.

What carries the argument

The key object is the face algebra $k\mathcal{F}$ of set compositions of $[n]$ — the monoid algebra of the set-composition monoid, whose elements are ordered set partitions of $[n]$. Through Bidigare's anti-isomorphism, the descent algebra is identified with the invariant subalgebra $(k\mathcal{F})^{S_n}$, so the shuffle $w_0 B_\alpha$ corresponds to the element $\widetilde{w}_0 e_{B_\alpha}$ in $k\mathcal{F}$. The paper defines the signed knapsack numbers $\operatorname{en}_\alpha(F) = (-1)^{n-\ell(F)} n_\alpha(F)$, where $n_\alpha(F)$ counts the faces of type $\alpha$ containing $F$. The decisive mechanism is the knapsack filtration of $k\mathcal{F}$: ordering the faces by increasing lengt

What would settle it

Compute the action of $w_0 A$ on the group algebra $\mathbb{Q}[S_n]$ for $n=5$ (a $120 \times 120$ integer matrix) and verify that its minimal polynomial equals $\prod_{k \in \{-3,-1,0,1,2,5\}}(x-k)$ with no repeated factors; a single missing root, an extra root, or a repeated factor would refute Theorems 8.1 and 9.5. Equivalently, check whether the operator is diagonalizable over $\mathbb{Q}$ by comparing the minimal polynomial's degree with the number of distinct eigenvalues.

Watch

Extended reading notes

Core claim

The central claim is Theorem 8.1: over any field of characteristic 0, for any composition $\alpha$ of $n$, the minimal polynomial of $w_0 B_\alpha$ (and its conjugate $B_\alpha w_0$) is $\prod_{k \in \operatorname{en}_\alpha(\mathcal{F})} (x-k)$, where $\operatorname{en}_\alpha(F) = (-1)^{n-\ell(F)} n_\alpha(F)$ is the signed knapsack number of the set composition $F$, with $n_\alpha(F)$ counting the number of set compositions of type $\alpha$ that contain $F$, and the product ranging over the distinct such numbers as $F$ varies over all set compositions of $[n]$. In the special case $\alpha = (1,n-1)$, which corresponds via the antipode to the top-to-random shuffle, this reduces to the expl

Load-bearing premise

The proof's triangular-matrix argument requires the strict knapsack monotonicity: for every face $G$ of type $\alpha$ that does not contain $F$, the count $n_\alpha(FG)$ must be strictly larger than $n_\alpha(F)$, and the same strictness must hold after averaging the counts with any nonnegative coefficient vector; if such an inequality ever degenerated to equality, the diagonal entries of the representing matrix would no longer be the signed knapsack numbers and the minimal-p

Editorial extensions

If this is right

For the reverse top-to-random shuffle, the full eigenvalue set is $L(n) = \{-n+2\} \cup [-n+4, n-3] \cup \{0\} \cup \{n\}$, so the operator is diagonalizable over $\mathbb{Q}$ with all eigenvalues integer.
The same conclusion holds for every $w_0 B_\gamma$ with nonnegative coefficients $\gamma_\alpha$: diagonalizable over any characteristic-0 field, with eigenvalues the $\gamma$-weighted signed knapsack numbers (Theorem 10.5).
The annihilating polynomial $\prod_{k \in \operatorname{en}_\alpha(\mathcal{F})}(w_0 B_\alpha - k)$ remains valid in positive characteristic, though diagonalizability can fail (explicitly, $x(x-1)^2$ for $n=4$, $p=3$).
Since conjugation by $w_0$ permutes the $B$-basis (Corollary 3.15), the formula applies equally to $B_\alpha w_0$ and $w_0 B_\alpha$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not pursued in the paper: the eigenvalue set for $\alpha=(1,n-1)$ has spectral gap 1 (the smallest nonzero absolute eigenvalue), so a standard comparison argument would likely give a $\Theta(n \log n)$ mixing time for the reversed shuffle, matching the ordinary top-to-random shuffle; the paper stops at the spectral description.
Not pursued in the paper: the same length-filtration triangularization that yields the minimal polynomial also gives a natural candidate for the algebraic multiplicities of each eigenvalue — the number of faces with that signed knapsack number — which would determine the full spectral decomposition on $\mathbb{Q}[S_n]$.
Not pursued in the paper: because the signed knapsack numbers can be negative, $w_0 B_\alpha$ is not a stochastic matrix; however, applying the results to a suitable positive combination or signed variant could still yield bounds for related Markov chains, since diagonalizability with real eigenvalues is the main structural input.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is real: for any composition α, the minimal polynomial of w0Bα (and Bαw0) over a characteristic-0 field is a product of distinct linear factors indexed by signed knapsack numbers. This genuinely extends Brown and Bidigare, who handled Bα and nonnegative combinations but not w0-composition. The reverse top-to-random case reduces to a clean integer set, which is the kind of result people in shuffle spectra will care about. The paper is carefully written and the central proof is complete. The knapsack filtration, the "more blocks" triangular-matrix argument in Lemma 8.3, and the alternating sum identities are all present and coherent. I also appreciate that the authors prove several background results self-contained in the appendix rather than just citing them. Soft spots are minor. Section 10 is explicitly an outline; the weighted generalization is plausible because the key strict monotonicity (Prop 10.3) carries over, but the reader should not expect full detail there. Edge cases n=2,3 are left to the reader, and there is a cosmetic slip in calling the length-filtration expansion upper triangular when it is lower triangular. Neither affects the main argument. The paper is also honest about the positive-characteristic behavior, which is a good sign. No circularity: the signed knapsack numbers are defined independently and then shown to be diagonal entries of a triangular matrix. This is a serious, well-grounded paper for descent algebra and shuffle spectral theory. I would cite it if I worked in that area, and it deserves a normal refereeing process rather than a desk rejection. My recommendation: send to peer review, accept after minor revisions.

Referee Report

0 major / 4 minor

Summary. The manuscript studies the elements w0Bα and Bαw0 in Solomon's descent algebra, where Bα are the B-basis elements and w0 is the longest permutation. The main theorem (Theorem 8.1) states that over a field of characteristic 0 the minimal polynomial of w0Bα is the product ∏_{k∈\widetilde{n}_α(F)}(x−k) over the distinct signed knapsack numbers \widetilde{n}_α(F)=(−1)^{n−ℓ(F)}n_α(F). The proof passes through Bidigare's anti-isomorphism between D(S_n) and the invariant subalgebra of the face algebra, constructs a knapsack filtration to obtain an annihilating polynomial (Theorem 7.2), and then uses a length filtration (Lemma 8.3) to show that right multiplication by e_{w0}e_{Bα} is lower-triangular with signed knapsack numbers on the diagonal, yielding diagonalizability and the exact minimal polynomial. Specializing to α=(1,n−1) gives Theorem 9.5: the reverse top-to-random shuffle has minimal polynomial ∏_{k∈L(n)}(x−k), with L(n)={−n+2}∪[−n+4,n−3]∪{0}∪{n}. Section 10 extends the result to nonnegative linear combinations Bγ=∑γαBα, presented as an explicit proof outline.

Significance. If the result holds, the paper provides a new family of diagonalizable shuffles with explicit integer spectra, extending the Bidigare–Brown framework in a direction that requires genuinely new arguments because w0 is not a face-monoid element. The central proof is detailed and largely self-contained, including a full proof of the Bidigare anti-isomorphism in Appendix A.3. The eigenvalue set is derived rather than fitted: signed knapsack numbers are defined before they are shown to be eigenvalues, and the minimal polynomial is identified by the triangular-matrix argument rather than by observation. The stress-test concern about Proposition 4.26 does not land: the proof that Nα(F) is a proper subset of Nα(FG) is valid and supplies exactly the strict monotonicity needed. Overall this is a solid contribution to the spectral analysis of shuffles associated with the descent algebra.

minor comments (4)

[Section 10, Theorems 10.5–10.6] These are stated as formal theorems but the Section 8 analogues (weighted triangular-matrix lemma and minimal-polynomial identification) are only sketched. If they remain formal claims, the missing details should be supplied or the section should be explicitly labeled as a proof sketch/remark. Additionally, Theorem 10.5 should repeat the hypothesis that k is ordered and contains the γα; 'field of characteristic 0' alone is not sufficient for the proof as written.
[Proposition 9.4 / Theorem 9.5] The proof of Proposition 9.4 assumes n>3 and leaves n=2,3 to the reader, even though Theorem 9.5 is stated for all n>1. Since these finite cases are part of the theorem, please insert the short checks or restrict the theorem to n>3 with a separate small-n statement.
[Sections 7–9, notation] The notation \widetilde{n}_α(F) for the value at a face and \widetilde{n}_α(F) for the set of all such values differs only typographically (by a thin space). This is easy to confuse, especially in displayed products over k∈\widetilde{n}_α(F). Consider using separate symbols, e.g., \operatorname{im}\widetilde{n}_α for the image set.
[Section 1, references] The introduction attributes the nonnegative-linear-combination diagonalizability result to 'Brown [23, Theorem 4.1]', but [23] is Schocker and Section 10 gives the correct attribution as Brown [4, Theorem 5] and Schocker [23, Theorem 4.1]. Please harmonize the citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the minimal-polynomial derivation is self-contained and not a disguised fit.

full rationale

The paper derives the minimal polynomial of w0Bα as a product over signed knapsack numbers. The signed knapsack numbers are defined in Definition 7.1 before any eigenvalue claim, and the proof does not fit them to the operator. Lemma 8.3 computes F·ew0eBα = enα(F) F + (longer faces) using the alternating-sum identity Lemma 6.2, which itself is proved from Lemma 6.1 via a bijection; neither identity assumes the target theorem. The operator is then triangular by length (Lemma 8.3), so its diagonal entries are the signed knapsack numbers; Theorem 7.2 supplies the squarefree annihilating polynomial via the knapsack filtration, with Proposition 4.26 proven by an explicit proper-subset argument. The passage from w0Bα to ew0eBα uses Bidigare's anti-isomorphism, for which the paper gives a self-contained proof in Appendix A.3 rather than relying solely on citation. Citations to the authors' own work are to ancillary or background material (e.g., [9], [12]) and are not load-bearing. Section 10 is explicitly an outline, but it is outside the main theorem. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed. The central claim therefore has independent mathematical content.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper is a pure derivation from standard algebraic combinatorics. There are no fitted numerical parameters, no empirically tuned constants, and no newly postulated physical objects. The only assumptions are algebraic: characteristic-0 fields for diagonalizability and nonnegative coefficients for the weighted extension. The signed knapsack number is a definition whose eigenvalue role is proved, not assumed.

assumptions (3)

standard math Finite-dimensional linear algebra: the minimal polynomial of an endomorphism divides any annihilating polynomial, and splitting into distinct linear factors over a field is equivalent to diagonalizability.
Used in Section 8 to pass from an annihilating product of distinct linear factors to diagonalizability and to identify the minimal polynomial.
domain assumption The base field has characteristic 0 for the main minimal polynomial and diagonalizability theorems.
Assumed explicitly in Theorem 8.1 and Theorem 9.5; the paper notes that positive characteristic can change the minimal polynomial, as with n=4 over F3.
domain assumption For the weighted generalization in Section 10, the coefficients γα are nonnegative and the base ring is ordered and contains them.
Nonnegativity is used in the weighted strict monotonicity Proposition 10.3 and in the filtration argument; the paper states that diagonalizability can fail without it.

invented entities (1)

signed knapsack number enα(F) = (-1)^(n - ℓ(F)) nα(F)
purpose: Labels the distinct eigenvalues of w0Bα and w0Bγ in the main spectral theorems.
This is a mathematical definition, not a physical postulate. Its role is established internally by the triangular-matrix proof, so it carries no outside falsifiable handle and creates no explanatory debt.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Top to random and reverse: analysis of a new descent algebra shuffle." pith.science (2026). https://pith.science/paper/6265EULQ

@misc{pith2026250806740,
  author       = {Pith},
  title        = {Pith review of: Top to random and reverse: analysis of a new descent algebra shuffle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6265EULQ}},
  note         = {Machine review of arXiv:2508.06740}
}

abstract

We study the "top-to-random-and-reverse shuffle", defined as the top-to-random shuffle in the symmetric group algebra composed with the permutation $w_0$ (which sends each $i$ to $n+1-i$). More generally, we analyze the composition of any B-basis element of the descent algebra with $w_0$. We show that the minimal polynomial of any such composition (over $\mathbb{Q}$) factors into distinct linear factors, which correspond to the "signed knapsack numbers" of set compositions. This is a counterpart to an analogous property of the B-basis elements themselves, which was proved by Brown using Bidigare's face monoid. In the case of the top-to-random-and-reverse shuffle, the minimal polynomial turns out to be $\prod_{k \in \set{-n+2} \cup \interval{-n+4, n-3} \cup \set{0} \cup \set{n}} \tup{x-k}$.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

27 extracted references · 25 canonical work pages

[1]

Hyperplane arrangement face algebras and their associated Markov chains

Patrick Bidigare. “Hyperplane arrangement face algebras and their associated Markov chains”. PhD thesis. Univ. Michigan, 1997

work page 1997
[2]

Noncommutative character theory of the symmetric group

Dieter Blessenohl and Manfred Schocker. Noncommutative character theory of the symmetric group. River Edge, NJ: World Scientific, 2005. isbn: 1-86094-511-2

work page 2005
[3]

Semigroup and ring theoretical methods in probability

Kenneth S. Brown. “Semigroup and ring theoretical methods in probability”. In: Representations of finite dimensional algebras and related topics in Lie theory and geom- etry. Proceedings from the 10th international conference on algebras and related topics, ICRA X, Toronto, Canada, July 15–August 10, 2002. Providence, RI: American Math- ematical Society (...

work page 2002
[4]

Semigroups, rings, and Markov chains

Kenneth S. Brown. “Semigroups, rings, and Markov chains”. In: J. Theor. Probab. 13.3 (2000), pp. 871–938. issn: 0894-9840. doi: 10.1023/A:1007822931408

work page doi:10.1023/a:1007822931408 2000
[5]

Mathematical developments from the analysis of riffle shuffling

Persi Diaconis. “Mathematical developments from the analysis of riffle shuffling”. English. In: Groups, combinatorics and geometry. Proceedings of the L. M. S. Durham symposium, Durham, UK, July 16–26, 2001. https://diaconis.ckirby.su.domains/ papers.html. River Edge, NJ: World Scientific,2003, pp. 73–97. isbn: 981-238-312-3

work page 2001
[6]

The mathematics of shuffling cards

Persi Diaconis and Jason Fulman. The mathematics of shuffling cards . English. Prov- idence, RI: American Mathematical Society (AMS), 2023. isbn: 978-1-4704-6303-8; 978-1-4704-7290-0. doi: 10.1090/mbk/146

work page doi:10.1090/mbk/146 2023
[7]

Generating a random permutation with random transpositions

Persi Diaconis and Mehrdad Shahshahani. “Generating a random permutation with random transpositions”. In: Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und ver- wandte Gebiete 57.2 (1981), pp. 159–179

work page 1981
[8]

Spectral analysis of random-to-random Markov chains

Antonius B Dieker and Franco V Saliola. “Spectral analysis of random-to-random Markov chains”. In: Advances in Mathematics 323 (2018), pp. 427–485. doi: 10.1016/ j.aim.2017.10.034

work page 2018

Show all 27 references

[9]

An Introduction to Algebraic Combinatorics

Darij Grinberg. An Introduction to Algebraic Combinatorics. 2025. arXiv: 2506.00738v1 [math.CO]. url: https://arxiv.org/abs/2506.00738v1

2025 arXiv
[10]

Is this sum of cycles invertible in QSn? MathOverflow

Darij Grinberg. Is this sum of cycles invertible in QSn? MathOverflow. version: 2025- 05-05. eprint: https://mathoverflow.net/q/308600. url: https://mathoverflow. net/q/308600

2025
[11]

The one-sided cycle shuffles, and other mysteries and wonders of the symmetric group algebra [talk slides]

Darij Grinberg. The one-sided cycle shuffles, and other mysteries and wonders of the symmetric group algebra [talk slides] . 2025. url: https : / / www . cip . ifi . lmu . de / ~grinberg/algebra/dc2023.pdf

2025
[12]

The one-sided cycle shuffles in the symmet- ric group algebra

Darij Grinberg and Nadia Lafreni `ere. “The one-sided cycle shuffles in the symmet- ric group algebra”. In: arXiv preprint arXiv:2212.06274 (2022). url: https://arxiv. org/abs/2212.06274

2022 arXiv
[13]

Vassilieva

Darij Grinberg and Ekaterina A. Vassilieva. The enriched q-monomial basis of the qua- sisymmetric functions. 2024. arXiv: 2309.01118v3 [math.CO]. url: https://arxiv. org/abs/2309.01118v3

2024 arXiv
[14]

A semigroup approach to wreath-product extensions of Solomon’s descent algebras

Samuel K. Hsiao. “A semigroup approach to wreath-product extensions of Solomon’s descent algebras”. In: Electron. J. Comb. 16.1 (2009), research paper r21, 9. issn: 1077-

2009
[15]

The miracle of integer eigenvalues

Richard Kenyon et al. “The miracle of integer eigenvalues”. English. In: Funct. Anal. Appl. 58.2 (2024), pp. 182–194. issn: 0016-2663. doi: 10.1134/S0016266324020072

2024 doi
[16]

Noncommutative symmetric functions. II: Transformations of alphabets

Daniel Krob, Bernard Leclerc, and Jean-Yves Thibon. “Noncommutative symmetric functions. II: Transformations of alphabets”. In: Int. J. Algebra Comput. 7.2 (1997), pp. 181–264. issn: 0218-1967. doi: 10.1142/S0218196797000113

1997 doi
[17]

A tour of representation theory

Martin Lorenz. A tour of representation theory . English. Vol. 193. Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2018. isbn: 978-1-4704- 3680-3; 978-1-4704-4905-6. doi: 10.1090/gsm/193. 56 Grinberg and Parlett Top to Random and Reverse August 8, 2025

2018 doi
[18]

On the matrix occurring in a linear search problem

Ravindra M. Phatarfod. “On the matrix occurring in a linear search problem”. In: Journal of Applied Probability 28.2 (1991), pp. 336–346. issn: 0021-9002. doi: 10.1017/ s0021900200039723

1991
[19]

Spectra of symmetrized shuf- fling operators

Victor Reiner, Franco Saliola, and Volkmar Welker. “Spectra of symmetrized shuf- fling operators”. In: Memoirs of the American Mathematical Society 228.1072 (2014). Preprint available at arXiv:1102.2460v2, pp. vi+109. issn: 0065-9266

2014 arXiv
[20]

Iterated-Integral Signatures in Machine Learning

Jeremy Francis Reizenstein. “Iterated-Integral Signatures in Machine Learning”. PhD thesis. University of Warwick, 2019, pp. ix+ 107. url: http://wrap.warwick. ac.uk/131162/

2019
[21]

Bruce E. Sagan. Combinatorics: The Art of Counting . Vol. 210. Graduate studies in mathematics. American Mathematical Society, 2020

2020
[22]

Hyperplane arrangements and descent algebras

Franco Saliola. Hyperplane arrangements and descent algebras . corrections at https: //www.cip.ifi.lmu.de/ ~grinberg/algebra/saliola-errata.pdf . url: https:// saliola.github.io/maths/publications/LectureNotes/DesAlgLectureNotes. pdf
[23]

The descent algebra of the symmetric group

Manfred Schocker. “The descent algebra of the symmetric group”. In: Represen- tations of finite dimensional algebras and related topics in Lie theory and geometry. Pro- ceedings from the 10th international conference on algebras and related topics, ICRA X, Toronto, Canada, Jul...

2002
[24]

A Mackey formula in the group ring of a Coxeter group

Louis Solomon. “A Mackey formula in the group ring of a Coxeter group”. In: Journal of Algebra 41 (1976), pp. 255–264

1976
[25]

Richard P . Stanley. Enumerative combinatorics. 2nd ed. Vol. 1. Cambridge University Press, 2011

2011
[26]

Nolan R. Wallach. Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals. Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 1...

1986
[8926]

url: https://eudml.org/doc/117303

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Hyperplane arrangement face algebras and their associated Markov chains

Patrick Bidigare. “Hyperplane arrangement face algebras and their associated Markov chains”. PhD thesis. Univ. Michigan, 1997

work page 1997

[2] [2]

Noncommutative character theory of the symmetric group

Dieter Blessenohl and Manfred Schocker. Noncommutative character theory of the symmetric group. River Edge, NJ: World Scientific, 2005. isbn: 1-86094-511-2

work page 2005

[3] [3]

Semigroup and ring theoretical methods in probability

Kenneth S. Brown. “Semigroup and ring theoretical methods in probability”. In: Representations of finite dimensional algebras and related topics in Lie theory and geom- etry. Proceedings from the 10th international conference on algebras and related topics, ICRA X, Toronto, Canada, July 15–August 10, 2002. Providence, RI: American Math- ematical Society (...

work page 2002

[4] [4]

Semigroups, rings, and Markov chains

Kenneth S. Brown. “Semigroups, rings, and Markov chains”. In: J. Theor. Probab. 13.3 (2000), pp. 871–938. issn: 0894-9840. doi: 10.1023/A:1007822931408

work page doi:10.1023/a:1007822931408 2000

[5] [5]

Mathematical developments from the analysis of riffle shuffling

Persi Diaconis. “Mathematical developments from the analysis of riffle shuffling”. English. In: Groups, combinatorics and geometry. Proceedings of the L. M. S. Durham symposium, Durham, UK, July 16–26, 2001. https://diaconis.ckirby.su.domains/ papers.html. River Edge, NJ: World Scientific,2003, pp. 73–97. isbn: 981-238-312-3

work page 2001

[6] [6]

The mathematics of shuffling cards

Persi Diaconis and Jason Fulman. The mathematics of shuffling cards . English. Prov- idence, RI: American Mathematical Society (AMS), 2023. isbn: 978-1-4704-6303-8; 978-1-4704-7290-0. doi: 10.1090/mbk/146

work page doi:10.1090/mbk/146 2023

[7] [7]

Generating a random permutation with random transpositions

Persi Diaconis and Mehrdad Shahshahani. “Generating a random permutation with random transpositions”. In: Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und ver- wandte Gebiete 57.2 (1981), pp. 159–179

work page 1981

[8] [8]

Spectral analysis of random-to-random Markov chains

Antonius B Dieker and Franco V Saliola. “Spectral analysis of random-to-random Markov chains”. In: Advances in Mathematics 323 (2018), pp. 427–485. doi: 10.1016/ j.aim.2017.10.034

work page 2018