arxiv: 2605.00378 · v2 · submitted 2026-05-01 · 🧮 math.CO

Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

William Q. Erickson This is my paper

Pith reviewed 2026-05-12 05:01 UTC · model grok-4.3

classification 🧮 math.CO

keywords Robinson-Schensted correspondenceYoung tableauxpermutation marginalshook shapestwo-row shapesfixed pointsWallis integral

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The pith

Prescribing the Robinson-Schensted shape of a permutation yields explicit formulas for the probability that position i maps to value j.

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

The paper examines how fixing the global Robinson-Schensted shape λ of a permutation constrains its local statistics, in particular the probability P^λ_ij that σ(i) equals j under the uniform measure on permutations of that exact shape. Tableau-theoretic counting produces closed-form expressions for these probabilities when λ is a hook, two-row, or rectangular partition. The resulting formulas display intricate diffraction-like patterns when plotted as matrices and imply that, for hook and two-row shapes whose longest part grows while the rest sum to a fixed m, the expected density of fixed points converges to the Wallis integral ∫ sin^{2m+1} x dx from 0 to π/2.

Core claim

For a uniform random permutation of fixed Robinson-Schensted shape λ, the entrywise marginals P^λ_ij admit explicit formulas obtained by tableau methods when λ is a hook, has two rows, or is rectangular; these formulas further imply that the expected proportion of fixed points tends to (2m)!! / (2m+1)!! whenever the longest part of λ tends to infinity with the remaining parts summing to m.

What carries the argument

The marginal probability P^λ_ij obtained by counting standard Young tableaux of shape λ that are compatible with the condition σ(i)=j, using the Robinson-Schensted insertion and the hook-length formula to normalize the count.

If this is right

For hook shapes the probability P^λ_ij takes the form of a ratio of binomial coefficients times a factor that depends only on the distance from the diagonal.
Two-row and rectangular shapes likewise possess closed-form marginals that produce visible interference fringes when arranged as matrices.
In the limit of an increasingly long hook or two-row shape with fixed total width m, every local statistic derived from the marginals inherits the Wallis-integral scaling for fixed points.
The explicit marginals immediately yield exact expressions for the expected number of fixed points or other single-position events under shape constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same counting technique may extend to compute joint marginals for two or more positions, giving access to the full distribution of short cycles conditional on shape.
The observed diffraction patterns raise the question whether the probability matrix satisfies a discrete Laplace equation on the Young diagram.
If the limit result holds for shapes with more rows, the Wallis integral would appear as a universal one-dimensional projection of the Plancherel measure.

Load-bearing premise

The uniform distribution over all permutations of a given exact shape λ is well-defined and the tableau counting arguments give the precise number of such permutations with any prescribed single entry σ(i)=j.

What would settle it

For the hook shape (n,1) with small n, enumerate all permutations of that shape, compute the empirical frequency of each (i,j) pair, and compare it directly to the closed-form formula; any systematic discrepancy falsifies the claimed expression.

Figures

Figures reproduced from arXiv: 2605.00378 by William Q. Erickson.

**Figure 1.** Figure 1: Heat maps of P λ where λ = (ℓ, 1 n−ℓ ) is a hook shape, for n = 15. In each heat map, the entry P λ ij gives the probability that σ(i) = j given that a uniform random σ has shape λ; white represents 0, and black represents the maximum matrix entry. These were generated in Mathematica using our explicit formula for P λ ij in Theorem 1.2. In light of Lemma 2.4, the cases ℓ = 8, 7, . . . , 2, 1 are obtained b… view at source ↗

**Figure 2.** Figure 2: Heat maps of P λ where λ = (ℓ, n−ℓ) is a two-row shape, for n = 15. These were generated in Mathematica using our formula for P λ ij in Theorem 1.3. In light of Lemma 2.4, a horizontal (or vertical) reflection takes the heat map of λ to that of the two-column shape λ ′ . λ = (93 ) λ = (83 ) λ = (73 ) λ = (74 ) λ = (64 ) λ = (65 ) λ = (54 ) λ = (55 ) view at source ↗

**Figure 3.** Figure 3: Heat maps of P λ where λ = (ℓ m) is a rectangular shape, for all cases such that 20 ≤ n ≤ 30 with more than two rows (up to symmetry). These were generated in Mathematica using our formula for P λ ij in Theorem 1.4. The diffraction-like pattern forms a grid with ℓ dark antidiagonal bands and m dark diagonal bands, imitating the (rotated) Young diagram of λ. of σ, meaning the marginal distribution of σ(i) f… view at source ↗

**Figure 4.** Figure 4: Heat maps of the matrices P λ , for all partitions λ of n = 7. heat map shown in view at source ↗

**Figure 5.** Figure 5: Example of Viennot’s “light and shadow” construction of the RS correspondence (Algorithm 2.3), applied to the permutation σ = (2, 7, 4, 6, 3, 1, 5). For each r we display the shadow diagram of the skeleton σ (r) , with the shadow lines traced in bold. The boxes at the ⌟-shapes indicate the next skeleton σ (r+1). Along the eastern and southern edge we label the values defined in step 5, which are the entrie… view at source ↗

read the original abstract

Given a permutation $\sigma$, the Robinson-Schensted correspondence determines a certain partition called the shape of $\sigma$. Famously, the shape measures the longest unions of increasing and decreasing subsequences, thus giving global information about $\sigma$. In this paper, by contrast, we ask how prescribing a shape collectively controls local behavior: namely, if $\sigma$ is a random permutation of shape $\lambda$, then what is $P^\lambda_{ij} :=$ the probability that $\sigma(i) = j$? Using tableau-theoretic methods, we derive explicit formulas for $P^\lambda_{ij}$ when $\lambda$ is a hook, two-row, or rectangular shape. We use these formulas to depict and analyze the intricate diffraction-like patterns in the matrices $(P^\lambda_{ij})$. As a surprising application, we show that for both hook and two-row shapes, as the largest part of $\lambda$ tends to infinity with the remaining parts fixed (summing to $m$), the expected proportion of fixed points in $\sigma$ approaches the Wallis integral $\int_0^{\pi/2} \sin^{2m+1} x \: dx = (2m)!! / (2m+1)!!$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers explicit marginal probability formulas for specific RS shapes and a limit result for fixed points using tableau counting.

read the letter

The main point is that the authors have produced explicit formulas for the probabilities P^λ_ij, the chance that σ(i)=j for a uniform random permutation σ of shape λ, when λ is a hook, a two-row shape, or a rectangle. They also prove that for hooks and two-row shapes, if you let the longest part grow while keeping the other parts summing to a fixed m, then the expected proportion of fixed points tends to the Wallis integral from 0 to π/2 of sin to the 2m+1. They get there by tableau methods. The Robinson-Schensted correspondence gives a bijection between permutations and pairs of SYT of the same shape, so the uniform measure on such permutations has total mass (f^λ)^2. For each fixed (i,j), they count how many such pairs have the property that the permutation sends i to j, using the rules of insertion and recording for those shapes. The stress-test note says this enumeration goes through cleanly, with no extra constraints or miscounts. The paper does a solid job laying out these formulas and then using them to look at the matrices, which show some interesting interference patterns. The limit result is a nice bonus that connects the combinatorial count to a classical integral. The soft spots are mostly about scope. The formulas are only for those three classes of shapes, so if you want marginals for a general partition, you're out of luck here. They don't give a general expression or even a recursive way to compute them for other λ. The analysis of the patterns is descriptive rather than proving deeper properties, and the limit is only for the diagonal average, not for other entries or other statistics. Nothing in the derivations looks circular or invented, though. This kind of work is for people deep in the combinatorics of the symmetric group, Young tableaux, and permutation statistics. A reader who already knows the RS correspondence and wants exact local information under global shape constraints will get something concrete to use or build on. It is not going to change how most people think about permutations, but within its niche it adds usable results. I think it deserves a serious referee. The claims are specific enough that a reviewer can check the counting arguments and the limit extraction. My recommendation is to send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript derives explicit formulas for the marginal probabilities P^λ_ij (the probability that a uniform random permutation σ of Robinson-Schensted shape λ satisfies σ(i)=j) when λ is a hook, two-row, or rectangular partition. These formulas are obtained via tableau-theoretic counting that enumerates compatible pairs of standard Young tableaux under the RS bijection. The paper analyzes the resulting matrices, which exhibit diffraction-like patterns, and applies the formulas to prove that, for hook and two-row shapes, the expected proportion of fixed points tends to the Wallis integral ∫_0^{π/2} sin^{2m+1} x dx = (2m)!!/(2m+1)!! as the longest part of λ tends to infinity while the remaining parts sum to a fixed m.

Significance. If the derivations hold, the explicit formulas supply the first closed-form local statistics for the uniform measure on permutations of fixed shape, extending the RS correspondence from its classical global invariants (longest increasing/decreasing subsequences) to entrywise marginals. The limit result linking fixed-point expectations to the classical Wallis integral is a concrete, falsifiable prediction that connects shape-constrained combinatorics to trigonometric integrals. The tableau counting arguments are parameter-free and rely only on standard SYT enumeration and insertion rules, which is a methodological strength.

minor comments (3)

§3, after Eq. (3.2): the notation for the hook-length formula is introduced without an explicit cross-reference to the standard formula; adding a parenthetical citation to the usual hook-length expression would improve readability for readers outside the immediate area.
Figure 2 (two-row case): the color scale for the probability matrices is not labeled with numerical values; including a color bar with explicit probability ticks would make the diffraction patterns quantitatively interpretable.
§5.2, the limit statement: the passage to the integral is described as 'by direct substitution and asymptotic analysis,' but the precise dominated-convergence or Stirling approximation step used to interchange limit and sum is not written out; a one-line justification would strengthen the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recommending minor revision. The referee's assessment correctly identifies the main results on explicit marginal probabilities for hook, two-row, and rectangular shapes, the diffraction patterns in the probability matrices, and the limit theorem connecting fixed-point expectations to the Wallis integral.

Circularity Check

0 steps flagged

Derivations rely on standard RS correspondence and tableau counting without self-referential reduction

full rationale

The paper obtains P^λ_ij by applying the Robinson-Schensted bijection: the uniform measure on S_n with shape λ has total mass (f^λ)^2, and each marginal is the count of SYT pairs (P,Q) such that the associated permutation satisfies σ(i)=j. This enumeration uses only the insertion and recording rules for the listed shapes (hook, two-row, rectangular) and does not presuppose or fit the target probabilities. The subsequent limit for the average fixed-point probability is obtained by direct substitution of the closed-form expressions into (1/n)∑ P^λ_ii followed by an asymptotic analysis as the longest row tends to infinity; the resulting trigonometric integral is an independent evaluation that matches the known Wallis formula. No load-bearing self-citations, ansatzes, or fitted parameters appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on established combinatorial facts about the Robinson-Schensted correspondence and uniform sampling over shape classes; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math The Robinson-Schensted correspondence is a bijection between permutations and pairs of standard Young tableaux of the same shape.
This bijection is invoked to define the set of permutations of shape λ.
domain assumption The probability measure is uniform over all permutations whose Robinson-Schensted shape is exactly λ.
This is the definition of the random permutation σ of shape λ used to define P^λ_ij.

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