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arxiv: 2604.16255 · v1 · submitted 2026-04-17 · 🧮 math.CO · math.RT

Multisymmetric functions on eventually constant cyclic graphs

Radford Green , Cornell Holmes , Mee Seong Im This is my paper

Pith reviewed 2026-05-10 07:49 UTC · model grok-4.3

classification 🧮 math.CO math.RT
keywords eventually constant k-tuplesmultisymmetric polynomialscyclic digraphsgenerating functionsgraph enumerationrepresentation charactersnilpotent structures
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The pith

Eventually constant k-tuples of functions on disjoint sets admit explicit multisymmetric generating polynomials that also serve as representation characters.

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

The paper generalizes the known correspondence between rooted trees and eventually constant self-maps on a single set to the case of k-tuples of maps between k separate finite sets. It shows that the generating polynomials enumerating these k-tuples, weighted by the structure of their induced digraphs, are symmetric across each of the k groups of variables and can be rewritten as the character of a natural representation of the product of k general linear groups. The same approach yields cardinality formulas and extends without change to the broader families of eventually N-cyclic and λ-cyclic k-tuples. A reader would care because the construction supplies a direct combinatorial model for a set-theoretic version of the nilpotent cone and links iteration of functions, cyclic graphs, and algebraic invariants in one framework.

Core claim

By identifying each eventually constant k-tuple with the digraph it induces on the disjoint union of the k vertex sets, explicit formulas are obtained for the associated generating polynomials in k blocks of variables. These polynomials are multisymmetric and equal the character of a representation of the product of the k general linear groups. The cardinality of the set of all such k-tuples is thereby determined, and the same method produces analogous generating functions and cardinalities for the more general eventually N-cyclic and λ-cyclic k-tuples.

What carries the argument

Eventually constant k-tuples of functions, identified with the digraphs on which iterated composition reaches a constant map on each component.

If this is right

  • The total number of eventually constant k-tuples on given set sizes is obtained by evaluating the generating polynomial at all variables equal to 1.
  • The same polynomial construction supplies weighted counts for the generalized eventually N-cyclic and λ-cyclic k-tuples.
  • The multisymmetric polynomials can be interpreted directly as characters of representations of the product of general linear groups.
  • The identification with induced digraphs extends the classical rooted-tree enumeration to a setting of multiple interacting function components.

Where Pith is reading between the lines

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

  • The same stabilization condition may allow enumeration of fixed-point-free or periodic functional graphs with prescribed cycle lengths.
  • The representation-theoretic expression suggests possible links to orbit-counting problems in products of matrix algebras.
  • Cardinality formulas could be tested on small instances to verify consistency with known counts of functional graphs having a single attracting component.

Load-bearing premise

The k-tuples reach constant functions after finitely many iterations of composition, which allows them to be faithfully modeled by their induced digraphs.

What would settle it

For k=2 and vertex sets of size 2, enumerate every pair of functions, compute the sum of the weights of those that become constant after iteration, and test whether the result equals the value given by the claimed multisymmetric polynomial.

Figures

Figures reproduced from arXiv: 2604.16255 by Cornell Holmes, Mee Seong Im, Radford Green.

Figure 1
Figure 1. Figure 1: Left: example of an eventually constant pair. Right: example of an eventually constant pair with |X1| = 4 and |X2| = 5. h f x1 y1 g z1 y1 f x1 y2 f x2 y3 f x3 y4 f x4 y5 g g h z1 g h h z2 g g z3 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: example of an eventually constant triple. Right: example of an eventually constant triple with |X1| = 4, |X2| = 5, and |X3| = 3. (f, g, h) induces a subgraph D containing a unique 3-cycle around v1, f(v1), gf(v1). Removing the edges of this cycle again yields an in-forest of Γ rooted around this cycle. See [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: example of a 3-cyclic 2-tuple. Right: example of one eventu￾ally 3-cyclic 2-tuple for |X1| = 4, |X2| = 5. Secondly, we apply the same limit to the factors of the bracketed term. Since the cycle size N is fixed, we can pass the limit inside the finite product: limn→∞   N Y−1 j=1  1 − j n    cn = N Y−1 j=1  limn→∞  1 − j n nc = N Y−1 j=1 e −jc = e −c PN−1 j=1 j = e −c( N 2 ) . Multiplying the… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Example of a map with two 3-cycles for sets of sizes (m, n, 2). We partition X1 ∪ X2 ∪ X3 into two disjoint sets, one set corresponding to one 3-cycle (they are circled above) and another set corresponding to the other 3-cycle (they are boxed above). Right: Example of a map with one 6-cycle for sets of sizes (m, n, 2). to send x32 to a red solid circle, (n − ℓ) m−k choices to send a red solid circle … view at source ↗
read the original abstract

The study of spanning trees and related structures is central in graph theory, closely connected to understanding functions between finite sets. This paper generalizes the established relationship between rooted trees and eventually constant endomorphisms to a wider context including $k$-tuples of functions among $k$ disjoint vertex sets. We derive a weighted count of eventually constant $k$-tuples, which are characterized by their stabilization to constancy upon iterated composition. This construction is the set-theoretic analogue of the nilpotent cone and offers new insight into the combinatorial structure of cyclic digraphs. By identifying these $k$-tuples with their induced digraphs, we construct explicit formulas for their generating polynomials and analyze the cardinality of the set of eventually constant $k$-tuples. These polynomials are multisymmetric in $k$ sets of variables and can be re-expressed as the character of a representation of the product of general linear groups. We extend the ideas to the more general structures of eventually $N$-cyclic and $\lambda$-cyclic $k$-tuples, which we define and provide similar theorems for their generating functions and cardinality.

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.

Referee Report

0 major / 2 minor

Summary. The paper generalizes the relationship between rooted trees and eventually constant endomorphisms to k-tuples of functions on k disjoint vertex sets. It derives weighted counts of eventually constant k-tuples (characterized by stabilization under iterated composition) by identifying them with induced cyclic digraphs, constructs explicit formulas for the associated generating polynomials, shows these polynomials are multisymmetric in k blocks of variables, and re-expresses them as characters of representations of products of general linear groups. The framework is extended to eventually N-cyclic and λ-cyclic k-tuples with analogous generating-function and cardinality results.

Significance. If the central combinatorial reduction holds, the work supplies concrete enumerative tools and representation-theoretic interpretations for structures analogous to the nilpotent cone in the set-function setting. The explicit formulas and direct digraph identification strengthen links between enumerative combinatorics, multisymmetric functions, and algebraic group representations, with potential utility for further study of cyclic graphs and stabilization phenomena.

minor comments (2)
  1. [Abstract] The abstract packs multiple distinct claims (weighted counts, explicit formulas, multisymmetry, character equalities, and the two extensions) into a single paragraph; a slightly expanded abstract or a roadmap paragraph at the end of the introduction would improve readability.
  2. [Introduction] Notation for the generating polynomials (e.g., the precise weighting scheme and the k blocks of variables) is introduced in the abstract but should be restated with a short example for k=2 in the first section to anchor the later theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The referee's summary correctly identifies the central contributions: the generalization from rooted trees and eventually constant endomorphisms to k-tuples on disjoint vertex sets, the weighted enumeration via induced cyclic digraphs, the explicit generating polynomials, the proof of multisymmetry, the character interpretation under products of general linear groups, and the extensions to eventually N-cyclic and λ-cyclic k-tuples.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central construction identifies eventually constant k-tuples with induced cyclic digraphs via the iterated-composition stabilization condition, then directly derives explicit generating polynomials that are shown to be multisymmetric and equal to characters of GL representations. This combinatorial reduction and the extensions to N-cyclic and λ-cyclic cases are presented as explicit constructions without reducing to fitted parameters, self-citations as load-bearing premises, or ansatzes imported from prior author work. The abstract and skeptic analysis confirm the counts and polynomials arise as derived objects from the set-theoretic analogue of the nilpotent cone, with no equations or claims that loop back to their own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background from graph theory and representation theory together with the new definitions of eventually constant k-tuples; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Established relationship between rooted trees and eventually constant endomorphisms on finite sets
    The paper explicitly generalizes this known relationship.
  • domain assumption Functions between finite sets induce digraphs whose combinatorial properties can be studied via generating polynomials
    Used to identify k-tuples with induced digraphs.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

  1. [1]

    [CIK+25] Weixi Chen, Mee Seong Im, Mikhail Khovanov, Catherine Lillja, and Nicolas Rugo,Pairs of even- tually constant maps and nilpotent pairs, arXiv preprint arXiv:2512.03367, to appear in Lett. Math. Phys. (2025), 1–15. [CILR25] Weixi Chen, Mee Seong Im, Catherine Lillja, and Nicolas Rugo,Eventually constant maps for two sets and nilpotent pairs, arXiv...

    work page arXiv 2025

  2. [2]

    Fine and Israel N

    [FH58] Nathan J. Fine and Israel N. Herstein,The probability that a matrix be nilpotent, Illinois J. Math.2 (1958), 499–504. [FS58] Miroslav Fiedler and Jiˇ r´ ı Sedl´ aˇ cek,On w-bases of directed graphs (¨ uber Wurzelbasen von gerichteten Graphen), ˇCasopis Pˇ est. Mat.83(1958), 214–225. [FS09] Philippe Flajolet and Robert Sedgewick,Analytic combinatori...

    work page 1958

  3. [3]

    Graham, Donald E

    [GKP89] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik,Concrete mathematics, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989, A foundation for computer science. [GR01] Chris Godsil and Gordon Royle,Algebraic graph theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York,

    work page 1989

  4. [4]

    [HJ94] Roger A

    26 RADFORD GREEN, CORNELL HOLMES, AND MEE SEONG IM [Hai10] Mark Haiman,Notes on the Matrix-Tree theorem and Cayley’s tree enumerator, Combinatorics https://math.berkeley.edu/∼mhaiman/math172-spring10/matrixtree.pdf (2010), 1–7. [HJ94] Roger A. Horn and Charles R. Johnson,Topics in matrix analysis, Cambridge University Press, Cambridge, 1994, Corrected rep...

    work page 2010

  5. [5]

    [Lei21] Tom Leinster,The probability that an operator is nilpotent, Amer. Math. Monthly128(2021), no. 4, 371–375. [PP94] Igor Pak and Alexander Postnikov,Enumeration of spanning trees of graphs, preprint (1994), 1–18. [Sta11] Andrew Stacey,Comparative smootheology, Theory Appl. Categ.25(2011), No. 4, 64–117. [Sta12] Richard P. Stanley,Enumerative combinat...

    work page 2021

  6. [6]

    [Sta24] ,Enumerative Combinatorics. Vol. 2, second ed., Cambridge Studies in Advanced Mathe- matics, vol. 208, Cambridge University Press, Cambridge, 2024, With an appendix by Sergey Fomin. [Wes96] Douglas B. West,Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ,

    work page 2024