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arxiv: 2604.26650 · v1 · submitted 2026-04-29 · 🧮 math.GR · math.PR

Probabilistic results for monoids of order-preserving transformations

Yang An , Wen Ting Zhang This is my paper

Pith reviewed 2026-05-07 12:05 UTC · model grok-4.3

classification 🧮 math.GR math.PR
keywords alphamathcaldistributionorder-preservingtransformationsfollowshypergeometricleqslant
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The pith

For uniform random elements of the monoid PO_n, image size given domain size r follows hypergeometric H(n+r-1, n, r); for the injective submonoid POI_n, image size is always equal to domain size and unconditional image size follows H(2n, n, n).

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

Order-preserving partial transformations map a subset of {1,2,...,n} to itself while respecting the natural order: if i is less than j then the image of i is less than or equal to the image of j. The authors treat these maps as equally likely and study the random variable that records how many distinct outputs appear. When the domain size is fixed at r, the number of possible images follows the hypergeometric distribution H(n+r-1, n, r). This arises because choosing an ordered image of size k inside the n points, while ensuring the partial map can be completed in an order-preserving way, reduces to a classic sampling-without-replacement count. For the injective submonoid, every map is one-to-one, so image size always equals domain size and is therefore degenerate; the unconditional distribution of image size is then another hypergeometric H(2n, n, n). The paper also supplies closed-form expressions for the mean and variance in each case.

Core claim

Y_r(α) follows a hypergeometric distribution H(n+r-1,n,r) for α ∈ PO_n, while Y_r(α) is degenerate and Y(α) follows a hypergeometric distribution H(2n,n,n) for α ∈ POI_n.

Load-bearing premise

The probability measure is the uniform distribution on the elements of PO_n (or POI_n), and the conditioning on domain size is with respect to this measure; the counting arguments that produce the hypergeometric parameters must hold without hidden restrictions on the partial maps.

read the original abstract

Let $\mathcal{PO}_n$ be the monoid of all order-preserving partial transformations on $X_n=\{1,\dots, n\}$ with the natural order, and let $\mathcal{O}_n$ and $\mathcal{POI}_n$ denote its submonoids of order-preserving full and injective partial transformations, respectively. For each transformation $\alpha\in\mathcal{PO}_n$, write the random variables $Y(\alpha)=|{\im}\alpha|$ and $Y_r(\alpha)=|{\im}\alpha|$ given that $|{\dom}\alpha|=r$ for $0 \leqslant r \leqslant n$. We determine the probability distribution, expectation and variance of $Y_r$ and $Y$ for $\mathcal{PO}_n$ and $\mathcal{POI}_n$. In particular, $Y_r(\alpha)$ follows a hypergeometric distribution $H(n+r-1,n,r)$ for $\alpha \in \mathcal{PO}_n$, while $Y_r(\alpha)$ is degenerate and $Y(\alpha)$ follows a hypergeometric distribution $H(2n,n,n)$ for $\alpha \in \mathcal{POI}_n$.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; no free parameters are fitted and no new entities are postulated. The sole domain assumption is uniformity of the probability measure on the monoid.

axioms (1)
  • domain assumption Uniform probability measure on the monoid PO_n (respectively POI_n)
    Required to define the random variables Y and Y_r as stated in the abstract.

pith-pipeline@v0.9.0 · 5495 in / 1307 out tokens · 75200 ms · 2026-05-07T12:05:52.980509+00:00 · methodology

discussion (0)

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