Rationality conjecture for submonoid generating functions

From papers

Let

G(x,y,z)=\sum_{\ell,m,n\geq 0}\#\operatorname{SubMon}([\ell-1]\times[m-1]\times[n-1])\frac{x^\ell y^m z^n}{\ell!m!n!}

be the multivariable exponential generating function for submonoids of products of finite commutative monoids. Rationality conjecture. The generating function $G(x,y,z)$ is a rational function in $e^x,e^y,e^z$ , and the corresponding higher-order multivariable generating function for $\#\operatorname{SubMon}(\prod_i [n_i-1])$ is likewise rational in the exponentials of its variables. This conjecture is motivated by the known one- and two-variable specializations and by partial information obtained from the eigenvalue expansion; the higher-dimensional rationality remains open.

Progress summary

Open

The conjecture remains open: existing work gives useful formulas in simpler settings, but no verified proof or counterexample has emerged for three or more variables.

The conjecture asserts that the trivariate exponential generating function $G(x,y,z)$ is rational in $e^x,e^y,e^z$ , with an analogous assertion in every higher dimension. The relevant preprint presents this as an unresolved extension of the known one- and two-variable theory.

Known results

For finite commutative idempotent monoids $M$ , transfer-matrix methods give $\#\operatorname{SubMon}(M\times[n])=\sum_{\lambda\in\Lambda}b_\lambda\lambda^n$ , with finite positive-integer eigenvalues and rational coefficients.
The two-variable product-of-chains case is related to poly-Bernoulli numbers; higher products have only partial eigenvalue information.

August 2025 preprint

The preprint “Enumerating submonoids of finite commutative monoids” states the conjecture as Conjecture 3.16 and records no proof or disproof. No independently corroborated recent resolution was found.

Current status (as of August 2026): The one- and two-variable cases have supporting formulas, but rationality for the trivariate and higher-dimensional generating functions remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young and Saron Zhu, “Enumerating submonoids of finite commutative monoids”, arXiv:2508.20786 (2025).

1 week ago

Counterexample: the trivariate generating function is not rational in the exponentials.

Let a_(l,m,n) be the number of submonoids of [l-1] x [m-1] x [n-1] under coordinatewise maximum, as in Conjecture 3.16 of arXiv:2508.20786. For n >= 2, consider the antichain

A_n = {(i,j,k) in {0,...,n-1}^3 : i+j+k=n-1}.

Its cardinality is binom(n+1,2). For each subset S of A_n, let M_S consist of the zero vector and all finite coordinatewise joins of members of S. This is a submonoid. The join of two distinct members of A_n has coordinate sum strictly greater than n-1, so M_S intersect A_n = S. Therefore all these submonoids are distinct and

a_(n,n,n) >= 2^binom(n+1,2).

Suppose, for contradiction, that G(x,y,z) = sum a_(l,m,n)x^l y^m z^n/(l!m!n!) were rational in e^x,e^y,e^z. Then there would be polynomials P,Q, with Q nonzero, satisfying

Q(e^x,e^y,e^z) G(x,y,z) = P(e^x,e^y,e^z)

as formal power series. Choose a positive integer L greater than every exponent appearing in Q, and substitute (x,y,z)=(t,Lt,L^2t). Distinct exponent triples of Q then give distinct integers i+Lj+L^2k, so q(t)=Q(e^t,e^(Lt),e^(L^2t)) is a nonzero analytic function. The resulting formal identity q(t)g(t)=p(t) implies that any zero of q at t=0 is canceled by p: hence g(t)=G(t,Lt,L^2t) is analytic near zero.

However all coefficients of g are nonnegative, and its diagonal term gives

[t^(3n)]g(t) >= L^(3n) 2^binom(n+1,2)/(n!)^3.

The right-hand side has unbounded (3n)-th root, so g has radius of convergence zero. This contradicts analyticity. Thus the trivariate rationality conjecture is false. The identical rank-layer argument refutes every higher-dimensional version with at least three variables; the known one- and two-variable rational formulas remain unaffected.

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Shivam Patel · 12 hours ago