Block-Restricted One-Swap Permutations

• The paper establishes the analytical tractability of block-restricted one-swap permutations, demonstrating exact finite-sample validity and variance reduction in both two-sample testing and combinatorial sorting.
• Block-restricted one-swap permutations are defined by permissible cross-block swaps within designated representatives, ensuring a rigorously constrained action space for efficient computation.
• The methodology leverages limited swap operations to yield tighter variance bounds and higher statistical power compared to full relabeling, enhancing both inference and sorting performance.

Block-restricted one-swap permutations are a highly structured class of permutations, relevant in both combinatorial sorting theory and the design of statistical permutation tests. The terminology spans two technical contexts: (1) structured test statistics in two-sample testing via block-restricted swaps between representatives, and (2) combinatorial characterization of permutations in the symmetric group that are sortable to the identity using exactly one prescribed block-interchange, as encoded by the context-directed swap (cds) operation. Both settings exhibit sharply restricted action spaces, admit exact analytical treatment, and find applications in statistical inference and permutation group theory.

1. Formal Definition of Block-Restricted One-Swap Permutations

In the statistical two-sample framework, suppose there are two groups $A = \{X_1, \ldots, X_{n_1}\}$ and $B = \{Y_1, \ldots, Y_{n_2}\}$, total $N = n_1 + n_2$ items. The set $\{1,2,\ldots,N\}$ is partitioned into $b$ disjoint blocks $\mathcal{B}_1, \ldots, \mathcal{B}_b$ via any label-invariant rule (e.g., quantiling on auxiliary covariates or kernel scores). A representative set $R \subset \{1, \ldots, N\}$ of size $|R| = \lfloor \rho N \rfloor$, $\rho \in (0,1]$, allocates quota per block. The permissible cross-swaps are

$$\mathcal{P} = \{ (i, j) : i \in A \cap R \cap \mathcal{B}_r,\, j \in B \cap R \cap \mathcal{B}_s,\, r \neq s \}.$$

A block-restricted one-swap permutation is any permutation obtainable from the identity via a single transposition in $\mathcal{P}$. More generally, an $L$-swap block-restricted path is a sequence of disjoint one-swap moves, with $L \leq |R|/2 = \rho N / 2$ (Ho, 29 Nov 2025).

In the combinatorial context, the cds sorting operation on $\mathfrak{S}_n$ acts by swapping two disjoint, specific blocks, determined by the occurrence pattern of pairs of “pointers” in the permutation’s pointer word. A permutation is block-restricted one-swap sortable if it requires exactly one cds operation to become the identity (Brown et al., 2020).

2. Algebraic and Combinatorial Structure

In the block-restricted two-sample setting, the set of admissible permutations forms a fixed subset $S_{\text{block}}$ of the symmetric group $S_N$, specified a priori by the blocks and representatives. These permutations are reachable via disjoint one-swap cross-block transpositions from the identity. This structure makes the analysis of test statistic increments and variance tractable, as the set structure is stable and independent of observed labels (Ho, 29 Nov 2025).

For cds one-swap-sortable permutations, there is an equivalent cycle-structure criterion: Define $X_n = (0\,1\,2\,\ldots\,n)$, $Y_\pi = (\pi(n)\,\pi(n-1)\,\ldots\,\pi(1)\,0)$, and $C_\pi = Y_\pi \circ X_n$. The permutation $\pi$ is cds one-swap-sortable if and only if $C_\pi$ has a unique $4$-cycle $(0\,n\,a\,b)$ for some distinct $a, b \in \{1,\ldots,n-1\}$ and all other cycles are fixed points. Equivalently, the strategic pile $\operatorname{SP}(\pi)$ has size two, and there is exactly one eligible pointer context (Brown et al., 2020).

3. Exact Validity and Increment Formulas in Statistical Testing

A principal advantage of block-restricted one-swap permutations in hypothesis testing is exact finite-sample validity. For any fixed (label-invariant) restricted set $S \subseteq S_N$, drawing i.i.d. permutations uniformly from $S$ and forming the usual permutation $p$-value yields type-I error control at all levels:

$$\mathbb{P}_{H_0}\{P \leq \alpha\} \leq \alpha,\;\, \forall \alpha \in [0,1].$$

When $S = S_\text{block}$ (all block-restricted one-swap permutations), randomization inference remains exact in finite samples without recourse to subgroup symmetry or worst-case conditions (Ho, 29 Nov 2025).

The restricted action space of single swaps enables closed-form increment calculations for common test statistics:

• Mean Difference: For $h = 1/n_1 + 1/n_2$, swapping $i \in A$, $j \in B$, $\Delta' - \Delta = h(Z_j - Z_i)$. Conditional variance under random one-swap is $h^2 \operatorname{Var}_w(Z_J - Z_I)$, contrasted with $O(h)$ variability under full-relabeling permutation.
• Unbiased $\widehat{\mathrm{MMD}}^2$: The one-swap change is $\psi_J^{B \to A} - \psi_I^{A \to B}$, and the conditional variance is the sum of within-class variances of the swap effectors.

The variance contraction (from $O(h)$ to $O(h^2)$) leads to tighter control of permutation quantiles and better Bernstein–Freedman tail proxies.

4. Power Enhancement and Critical Value Behavior

Under full relabeling, the null permutation distribution of the statistic $T$ admits Chebyshev or normal-approximation quantile bounds:

$$q_{1-\alpha}^{\mathrm{full}} \leq \mathbb{E}[T] + \sqrt{\operatorname{Var}_{\mathrm{full}}(T)/\alpha}.$$

The block-restricted scheme, by contrast, yields Bernstein–Freedman upper bounds:

$$q_{1-\alpha}^{\mathrm{rest}} \leq \mathbb{E}[T\,|\,S_N] + 2\sqrt{L v_\ast \log(1/\alpha)},$$

with $v_\ast = h^2 \operatorname{Var}_w(Z_J - Z_I)$ for the mean or $\tau_A^2+\tau_B^2$ for MMD, $L = \rho N/2$. Because $h \ll 1$, $q_{1-\alpha}^{\mathrm{rest}}$ is substantially below $q_{1-\alpha}^{\mathrm{full}}$ for the same $\alpha$, for large $n$. The result is higher statistical power under alternatives:

$$\beta^{\mathrm{rest}}(\delta) \approx 1 - \Phi\left(\frac{q_{1-\alpha}^{\mathrm{rest}} - \delta}{\sigma_{\mathrm{alt}}}\right),\quad \beta^{\mathrm{full}}(\delta) \approx 1 - \Phi\left(\frac{q_{1-\alpha}^{\mathrm{full}} - \delta}{\sigma_{\mathrm{alt}}}\right).$$

Pointwise, $$\beta^{\mathrm{rest}}(\delta) \geq \beta^{\mathrm{full}}(\delta)$$ (Ho, 29 Nov 2025). Empirically, gains are maximized for balanced, large samples and block schemes that exploit high-contrast structures (e.g., kernel-score stratification).

5. Enumeration and Structure in $\mathfrak{S}_n$

Within the cds framework, every one-swap-sortable permutation corresponds uniquely to an unordered pair $\{a, b\} \subset \{1,\ldots,n-1\}$, $a \neq b$, and a choice of orientation for the 4-cycle. The total number is $2\binom{n-1}{2} = (n-1)(n-2)$. These permutations are extremely sparse, $\Theta(n^2)$ in total against $n!$ overall. Detecting single-swap-sortability is $O(n)$: compute the strategic pile and scan for pointer-interleaving (Brown et al., 2020).

Such permutations sit at the minimal nontrivial end of the cds complexity spectrum: the first nonzero member in the species-by-contexts partition $M_{n, k}$ at $k=1$. They admit two equivalent block forms depending on the context’s order in the pointer word, e.g.,

$$[\; y+1, y+2, \dots, n;\;\; 1, 2, \dots, x;\;\; x+1, \dots, y\; ]$$

for pointer pairs $(x,x+1), (y, y+1)$ with $1 \leq x < y \leq n-1$.

6. Relationship to Broader Permutation and Testing Theory

Block-restricted one-swap permutations provide a test-bed for studying the trade-off between label-exchange symmetry and power in permutation tests. Their a priori definition ensures exact validity and sharp variance behavior, exploiting fixed group structure rather than average- or worst-case coverage.

In the cds sorting paradigm, block-restricted one-swap permutations characterize the boundary of “minimal non-sortability”—the fewest possible eligible pointer contexts and smallest possible strategic pile size (2) to require any block-interchange operation. This positions them as a combinatorial extreme, with implications for sorting complexity and group action enumeration.

These frameworks, found in "Restricted Block Permutation for Two-Sample Testing" (Ho, 29 Nov 2025) and "Classifying Permutations under Context-Directed Swaps and the cds game" (Brown et al., 2020), exemplify structured, low-complexity randomization mechanisms enabling rigorous finite-sample statistical guarantees and rich algebraic properties.