One-Sided Cycle Shuffles

• One-sided cycle shuffles are operations that reinsert a card only at or below its original position, enforcing a directional, lower-triangular constraint.
• They use linear combinations of cyclic operators whose triangularization yields a spectrum with a Fibonacci bound on the number of eigenvalues.
• These shuffles underpin models in permutation statistics and cryptography, offering insights into random walks and secure card-based protocols.

One-sided cycle shuffles are a family of shuffle constructions in which admissible moves are constrained by a preferred cyclic direction or by a one-sided insertion rule, rather than by the full symmetry of arbitrary interleavings or two-sided cycle actions. In the symmetric-group algebra, the term denotes linear combinations of the operators

$$t_i:=\mathrm{cyc}_{i}+\mathrm{cyc}_{i,i+1}+\mathrm{cyc}_{i,i+1,i+2}+\cdots+\mathrm{cyc}_{i,i+1,\ldots,n},$$

which encode removal of the card at position $i$ and reinsertion at a uniformly random position weakly below $i$ (Grinberg et al., 2022). In graph-based card cryptography, the same phrase is used for rotation-only shuffles on a directed cycle $\overrightarrow{C_n}$, where reflections are excluded and the automorphism group reduces to $\mathbb{Z}_n$ (Shinagawa et al., 2022). In algebraic combinatorics, related one-sided variants arise as anchored or restricted cyclic shuffles, especially through the subsets $[T] W_i [o]$ that refine the full cyclic shuffle set by constraining the interleaving to a single arc (Domagalski et al., 2021). The subject therefore spans representation theory, random walks on $S_n$, cyclic permutation statistics, and secure card-based computation.

1. Symmetric-group algebra formulation

In the most explicit algebraic usage, one-sided cycle shuffles are elements of the group algebra $\mathbf{k}[S_n]$ generated by the operators $t_1,\dots,t_n$, where each $t_i$ is the “somewhere-to-below” operator at level $i$0 (Grinberg et al., 2022). The underlying move is one-sided in the sense that the card currently in position $i$1 may only be reinserted at positions $i$2, never above $i$3. Equivalently,

$i$4

with $i$5.

A general one-sided cycle shuffle is a linear combination

$i$6

When $i$7, not all zero, the associated right random walk chooses a position $i$8 with probability proportional to $i$9, and then reinserts the selected card uniformly among the positions $i$0 (Grinberg et al., 2022). The Markov-normalized form associated with a probability distribution $i$1 on $i$2 is

$i$3

Several named shuffles occur as special cases. The operator $i$4 is the classical top-to-random shuffle. The random-to-below shuffle is

$i$5

The unweighted one-sided cycle shuffle corresponds to $i$6, which averages uniformly over all admissible pairs $i$7 with $i$8 (Grinberg et al., 2022).

This operator-theoretic meaning of the term is distinct from the usage in cyclic combinatorics, where “one-sided” refers to anchored cyclic interleavings, and from the usage in card-based cryptography, where it refers to rotation-only automorphisms of a directed cycle. The shared theme is the imposition of an orientation-sensitive or lower-triangular constraint.

2. Lacunar parametrization, triangularization, and spectrum

The spectral theory of one-sided cycle shuffles is governed by lacunar subsets. A subset $i$9 is lacunar if it contains no two consecutive integers. Writing

$\overrightarrow{C_n}$0

the integers

$\overrightarrow{C_n}$1

encode the distance from $\overrightarrow{C_n}$2 to the next lacunar “barrier” weakly to its right (Grinberg et al., 2022).

The key structural result is simultaneous triangularization of the right-multiplication operators $\overrightarrow{C_n}$3 on $\overrightarrow{C_n}$4. The paper constructs invariant spaces $\overrightarrow{C_n}$5, a Fibonacci filtration

$\overrightarrow{C_n}$6

and a “descent-destroying basis” $\overrightarrow{C_n}$7 indexed by permutations $\overrightarrow{C_n}$8, such that each $\overrightarrow{C_n}$9 acts upper triangularly with diagonal entries $\mathbb{Z}_n$0, where $\mathbb{Z}_n$1 runs through the lacunar subsets ordered by increasing sum (Grinberg et al., 2022). Concretely,

$\mathbb{Z}_n$2

For

$\mathbb{Z}_n$3

the spectrum is

$\mathbb{Z}_n$4

The operator satisfies the annihilating polynomial

$\mathbb{Z}_n$5

If the values $\mathbb{Z}_n$6 are pairwise distinct, then $\mathbb{Z}_n$7 is diagonalizable (Grinberg et al., 2022).

The number of possible eigenvalues is therefore at most $\mathbb{Z}_n$8, a Fibonacci bound. This is markedly smaller than $\mathbb{Z}_n$9, and it reflects the fact that the operators $[T] W_i [o]$0 do not commute in general, yet remain simultaneously triangularizable.

3. Multiplicities and representation theory

The multiplicities of eigenvalues in the regular representation admit explicit combinatorial formulas. If $[T] W_i [o]$1 is lacunar, with $[T] W_i [o]$2, $[T] W_i [o]$3, and $[T] W_i [o]$4, then the dimension increment of the corresponding graded piece is

$[T] W_i [o]$5

When the eigenvalues are distinct, these $[T] W_i [o]$6 are the algebraic multiplicities; in general, multiplicities are obtained by aggregating the $[T] W_i [o]$7 over coincident eigenvalues (Grinberg et al., 2022).

A later representation-theoretic development determines the action of one-sided cycle shuffles on every Specht module $[T] W_i [o]$8. For a lacunar set $[T] W_i [o]$9, let $S_n$0, set $S_n$1, and define $S_n$2. Then the multiplicity parameter

$S_n$3

is a Littlewood–Richardson coefficient (Grinberg, 1 Aug 2025). For

$S_n$4

the eigenvalues of the left action $S_n$5 on $S_n$6 are precisely the $S_n$7 for lacunar $S_n$8 with $S_n$9. In the generic case, the algebraic multiplicity of $\mathbf{k}[S_n]$0 is $\mathbf{k}[S_n]$1, and if these eigenvalues are pairwise distinct then $\mathbf{k}[S_n]$2 is diagonalizable (Grinberg, 1 Aug 2025).

This representation-theoretic description replaces tableau-content formulas by lacunar-set data and Littlewood–Richardson multiplicities. The operators $\mathbf{k}[S_n]$3 are therefore unlike Jucys–Murphy elements: they are not commuting content operators, but their spectra are still accessible through an explicit filtration.

4. Probabilistic interpretations and mixing behavior

As random walks on $\mathbf{k}[S_n]$4, one-sided cycle shuffles encode non-reversible, directionally constrained dynamics. The random-to-below chain is the Markov-normalized walk in which a card is chosen uniformly at random and reinserted uniformly below. For this chain, a strong stationary time is obtained by placing a bookmark right above the card that is initially at the bottom; each time a card above the bookmark is moved into the bookmarked gap, the bookmark moves up one position, and when it reaches the top, the next step completes a uniform random permutation (Grinberg et al., 2022). Its expectation satisfies

$\mathbf{k}[S_n]$5

A related but distinct family is the one-sided transposition shuffle. At each step, one chooses a “right-hand” position $\mathbf{k}[S_n]$6 uniformly from $\mathbf{k}[S_n]$7, then a “left-hand” position $\mathbf{k}[S_n]$8 uniformly from $\mathbf{k}[S_n]$9, and transposes the cards at positions $t_1,\dots,t_n$0 and $t_1,\dots,t_n$1 (Bate et al., 2019). Its probability operator is

$t_1,\dots,t_n$2

and it admits an explicit tableau-indexed spectrum through Jucys–Murphy elements. The chain has a total-variation cutoff at time $t_1,\dots,t_n$3 with an $t_1,\dots,t_n$4 window (Bate et al., 2019). A weighted version with weights $t_1,\dots,t_n$5 exhibits cutoff at time $t_1,\dots,t_n$6, recovering the classical $t_1,\dots,t_n$7 random-transposition scale at $t_1,\dots,t_n$8 (Bate et al., 2019).

Another one-sided family is the card-cyclic-to-random shuffle, in which cards are processed in deterministic cyclic order and reinserted uniformly. In the relabeling variant, cards are relabeled after each round according to their current positions. The mixing time remains of order $t_1,\dots,t_n$9 steps: $t_i$0 for absolute constants $t_i$1 (Jonasson, 2015). This indicates that deterministic cyclic scheduling of the moved card, by itself, does not accelerate mixing beyond the $t_i$2 scale.

A related misconception is that all one-sided shuffles should share the same spectral framework. The literature does not support that conclusion. Somewhere-to-below shuffles are analyzed through lacunar subsets and triangularization (Grinberg et al., 2022), one-sided transpositions through tableau contents and Jucys–Murphy elements (Bate et al., 2019), and cyclic-to-random insertion models through round-level single-card dynamics (Jonasson, 2015).

5. Cyclic and one-sided variants in algebraic combinatorics

In algebraic combinatorics, the phrase “one-sided cycle shuffle” is less standardized. Cyclic permutations are equivalence classes under rotation, not reflection. If $t_i$3, then

$t_i$4

The cyclic shuffle $t_i$5 consists of cyclic permutations $t_i$6 in which $t_i$7 and $t_i$8 occur as circular subwords (Domagalski et al., 2021).

The paper on cyclic shuffle compatibility introduces a refinement

$t_i$9

and proves that

$i$00

The term “one-sided cycle shuffles” does not appear explicitly there, but these restricted subsets are interpreted as one-sided constraints because the interleaving is confined to a single arc of the cycle (Domagalski et al., 2021). The lifting lemma of that paper operates piecewise on the sets $i$01, so cyclic descent-type statistics are controlled on each one-sided piece, not only on the full cyclic shuffle set.

A second strand concerns cyclic major index. A cyclic permutation $i$02 is represented by the canonical word $i$03 whose first letter is the largest element, and the cyclic major index is defined by

$i$04

The cyclic analogue of Stanley’s shuffle theorem gives a bivariate formula for cyclic shuffles by cyclic descent number and cyclic major index (Ji et al., 2022). That paper does not explicitly define “one-sided cycle shuffles” either, but it explains that its two-sided cyclic formula specializes directly to anchored one-sided settings by fixing the start $i$05 of one of the cycles and using the corresponding term involving $i$06. Thus, in this literature, one-sidedness means anchoring one cyclic factor or restricting interleaving to a chosen arc, rather than enforcing a somewhere-to-below move.

This suggests a broad terminological point: the phrase “one-sided cycle shuffles” is not globally uniform across arXiv literatures. In representation theory and shuffle algebras it denotes linear combinations of the $i$07; in cyclic permutation theory it is usually an interpretation or specialization of a more general cyclic shuffle construction (Grinberg et al., 2022).

6. Directed cycles, automorphisms, and card-based cryptography

In card-based cryptography, one-sided cycle shuffles arise from graph automorphisms. For a directed graph $i$08, the graph shuffle associated with $i$09 is the uniform closed shuffle over the induced vertex-action group $i$10. For the directed cycle $i$11, one has

$i$12

whereas the undirected cycle $i$13 has

$i$14

Accordingly, a one-sided cycle shuffle is the uniform closed shuffle over rotations only, with reflections excluded by the orientation of edges (Shinagawa et al., 2022).

This distinction is operationally important. Using a directed cycle enforces one-sidedness because reflections would invert edge orientations and thus are not automorphisms. For $i$15, the graph-shuffle protocol samples uniformly over $i$16, so the output is a uniformly random rotation of the input sequence (Shinagawa et al., 2022).

The improved pile-scramble implementation for a directed graph with $i$17 vertices and $i$18 edges uses $i$19 cards. For $i$20, where $i$21, this becomes $i$22 cards, improving on the earlier $i$23-card implementation (Shinagawa et al., 2022). In the special case $i$24, the protocol uses $i$25 cards rather than $i$26, and the effect is exactly a uniformly random rotation $i$27 of the input $i$28 (Shinagawa et al., 2022).

Security is formulated in the Mizuki–Shizuya model. The chosen automorphism $i$29 must be information-theoretically hidden from the visible trace. For directed cycles, the protocol samples uniformly over rotations, and because every vertex has outdegree $i$30, the internal $i$31 step is vacuous; the simplification does not affect the uniformity or the security argument (Shinagawa et al., 2022).

This cryptographic notion of one-sidedness is therefore geometric and group-theoretic: it means orientation-preserving cyclic action. It should not be conflated with the somewhere-to-below meaning in $i$32-module theory, although both exclude an opposite-direction symmetry.

7. Scope, related generalizations, and open directions

Several generalizations place one-sided cycle shuffles within wider shuffle theories. In complex reflection groups $i$33, the cyclotomic $i$34-shuffle

$i$35

models a move in which a distinguished card is removed, reinserted at any position, and simultaneously rotated by any of the $i$36 possible cyclic orientations (Ogievetsky et al., 2017). There, “one-sided” refers to moving a single distinguished card rather than interleaving two subdecks. The resulting left-multiplication operator has eigenvalues $i$37 for $i$38, and the associated random walk has spectral gap $i$39 for $i$40 (Ogievetsky et al., 2017).

Within the somewhere-to-below theory itself, several open directions are explicit. The 2022 paper identifies non-diagonalizable parameter choices when eigenvalues collide and emphasizes that simultaneous triangularization, rather than simultaneous diagonalization, is the correct general framework (Grinberg et al., 2022). The 2025 representation-theoretic sequel determines spectra on Specht modules but still through a filtration-based method rather than a tableau-content calculus, indicating that the algebra generated by the $i$41 is structurally different from more classical commutative families (Grinberg, 1 Aug 2025).

In cyclic combinatorics, the one-sided subsets $i$42 suggest refined structure constants and finer distributional questions for cyclic descent-type statistics, but the papers cited treat these subsets mainly as tools for lifting linear shuffle compatibility to the cyclic setting (Domagalski et al., 2021). In card-based cryptography, explicit open directions include reducing the number of cards or pile-scramble steps further for special graph families and extending beyond uniform closed shuffles over automorphism groups (Shinagawa et al., 2022).

Across these settings, the recurring invariant is asymmetry. One-sided cycle shuffles restrict a shuffle by orientation, anchoring, or insertion direction, and this asymmetry is precisely what makes their algebraic spectra, combinatorial statistics, and cryptographic realizations differ from their two-sided analogues.