Segre Characteristic Equivalence

Jessie Pitsillides

REVIEW 3 major objections 5 minor 6 references

Segre Characteristic Equivalence

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that, with eigenvalues treated as unlabeled, the Jordan block patterns of an $n\times n$ complex matrix are counted by the partition-of-partitions number $P(n)$.

desk verdict Correct classical enumeration, but the main theorem is false as stated, the proof is circular, and the appendix code overcounts. read the letter →

arxiv 2506.12065 v1 pith:YZJQJU5X submitted 2025-06-01 math.GM

Jessie Pitsillides This is my paper · ORCID

classification math.GM MSC 15A2105A17

keywords SegrecharacteristicJordannormalformpartitionofpartitionsrankpatternFerrersdiagrameigenvaluegroupingA001970matrixclassification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks a counting question with no eigenvalue data: for an $n\times n$ complex matrix, how many different Segre characteristics (Jordan block sizes grouped by eigenvalue) are possible? Its answer is the partition-of-partitions number $P(n)$, whose generating function is $\prod_{k\ge 1}(1-x^k)^{-p(k)}$ and whose first values are $1, 1, 3, 6, 14, 27, 58, \ldots$. Theorem 1 shows that a matrix's rank pattern determines its Segre characteristic, and Theorem 2 counts the possibilities by first partitioning $n$ into block sizes and then partitioning those sizes according to which eigenvalues share blocks. If the theorem is right, a classical 1854 enumeration sequence turns out to classify all possible Jordan structures of matrices of a given size.

What carries the argument

The load-bearing objects are the Segre characteristic, a list of integer partitions recording Jordan block sizes grouped by eigenvalue, and the rank pattern $R_A(\lambda) = \{\operatorname{rank}((A-\lambda I)^k) : k \ge 0\}$ for each eigenvalue. The paper uses the dictionary that the conjugate partition of the nullity growth sequence gives the block sizes for that eigenvalue, with Ferrers diagrams making the step concrete. The counting machinery is the partition-of-partitions function $P(n)$ with generating function $\prod_{k\ge 1}(1-x^k)^{-p(k)}$: for every integer partition of $n$, counting the ways to group its parts into nonempty subsets gives a partition of that partition, and summing over all starting partitions gives $P(n)$.

What would settle it

Enumerate directly all possible Jordan block patterns for $n=3$: the theorem predicts exactly six, namely $[(3)]$, $[(2,1)]$, $[(2),(1)]$, $[(1,1,1)]$, $[(1,1),(1)]$, and $[(1),(1),(1)]$. If any $3\times 3$ matrix over $\mathbb{C}$ has a Segre characteristic outside this list, the claimed formula is wrong.

Watch

Extended reading notes

Core claim

At the centre is Theorem 2: up to the paper's convention that eigenvalues are unlabeled slots, there are exactly $P(n)$ distinct Segre characteristics for matrices in $M(n,\mathbb{C})$, where $P(n)$ has generating function $\prod_{k\ge 1}(1-x^k)^{-p(k)}$. The proof decomposes a Jordan form into blocks whose sizes form an integer partition of $n$; choosing which block sizes belong to the same eigenvalue is itself a partition of each part, so the total count is the coefficient of $x^n$ in the product. The paper's Theorem 1 supplies the structural underpinning: for each eigenvalue, the sequence of ranks of powers of $A-\lambda I$ determines the nullity growth pattern, and taking the conjugate Ferrers diagram recovers the block sizes. On this reading, the finite number $P(n)$ is genuinely a property of the dimension alone.

Load-bearing premise

The entire count depends on treating eigenvalues as unnamed slots so that only the sizes and groupings of Jordan blocks matter; if the actual complex values of eigenvalues had to be counted, there would be infinitely many possibilities even for a $1\times 1$ matrix.

Editorial extensions

If this is right

For every $n$, the possible Jordan block patterns are in one-to-one correspondence with the partitions of partitions of $n$, so they can be listed algorithmically from integer partitions alone.
The counts for $n=3,4,5,6$ are $6, 14, 27, 58$, so the number of distinct Jordan structures grows faster than the ordinary partition count $p(n)$.
Because the rank pattern determines the Segre characteristic, computing ranks of powers of $A-\lambda I$ for each eigenvalue is enough to read off the Jordan block sizes, making the classification computable without first finding a Jordan basis.
Two matrices with the same Segre characteristic need not be similar, since the eigenvalues themselves may differ; Segre equivalence is a coarser invariant that ignores the values of the eigenvalues.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because Theorem 2 counts patterns rather than literal matrices, a charitable restatement is that the number of Segre characteristics (eigenvalue-unlabeled Jordan types) of dimension $n$ is $P(n)$; without unlabeled eigenvalues, every type contains infinitely many matrices.
The same argument carries over to any algebraically closed field, since the Jordan decomposition exists there and the eigenvalue values never enter the block-size count.
A labeled-eigenvalue version would be uncountable: for $n\ge 1$, $\lambda I$ with $\lambda$ varying over $\mathbb{C}$ gives infinitely many similarity classes, while the type count remains $P(n)$.
A direct check of the paper's algorithm is to generate all sorted multipartitions for $n=7$ and count $111$ of them, matching the published next term.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the intended count is right, but the main theorem as written is false and the proof is circular; the appendix code also overcounts.

The paper does contain a correct classical observation: the number of Segre characteristic types for n×n matrices, with eigenvalues treated as unlabeled placeholders, is the partitions-of-partitions number P(n) from OEIS A001970, originally due to Sylvester. The worked enumeration for n=4 is accurate, and the figures for n=4,5,6 match the sequence. So there is a real kernel here.

The problem is Theorem 2. It says there are P(n) Segre Characteristic equivalent matrices of dimension n. Literally that is false: for any Segre characteristic there are infinitely many matrices realizing it, since eigenvalue values are complex numbers. Scalar matrices λI alone give an infinite family for n=1, all with characteristic [(1)]. The finite count only works if you count equivalence classes where eigenvalues are treated formally, i.e., a Segre characteristic is an unordered multiset of integer partitions, and matrices differing only by the numeric values of eigenvalues are identified. That convention is never stated; it appears implicitly in the "given only the dimension" phrasing and in the proof.

The proof itself is circular: it says the count is "by definition the partitions of partitions of n", which is exactly the definition of P(n). It never proves that the block-grouping operation is actually a bijection to the set of partitions of partitions. So the central claim is a restatement, not a derivation.

There is also a concrete bug in the appendix. The Mathematica function P[n] returns 15 for n=4, not 14, because Multipartitions replaces each Jordan block size by a partition of that size, splitting blocks rather than grouping distinct blocks under a shared eigenvalue. The text's P(4)=14 is correct; the code does not implement that count.

Who is this for? Someone who wants a handy picture of the number of unlabeled Jordan types and a pointer to Sylvester's sequence could use the examples as a reference. As a proof contribution, it is not there. The fix is straightforward in principle: state the theorem as a count of Segre characteristic equivalence classes, make the unlabeled-eigenvalue convention explicit, and prove the bijection with partitions of partitions (or cite a full proof). The distinction between matrices and classes should be the headline.

I would send it to a referee rather than desk reject, because the kernel is correct and the needed corrections are local, not a change of subject. But I would not cite it as is.

Referee Report

3 major / 5 minor

Summary. The paper asks how many Segre-characteristic-equivalent matrices of dimension n exist, and answers that the number is P(n), the so-called partition-of-partitions count with generating function ∏_{k≥1}(1−x^k)^{-p(k)}, matching OEIS A001970. It defines Segre equivalence, reviews the rank-pattern criterion for Jordan block structure, states Theorem 1 (rank pattern determines the Segre characteristic), and states Theorem 2 (there are P(n) Segre Characteristic equivalent matrices of dimension n). The proof of Theorem 2 is one sentence: the count is 'by definition the partitions of partitions of n.' A worked example for n=4 lists 14 types, and the appendix gives Mathematica code to draw the Jordan block patterns for small n.

Significance. If the theorem is interpreted as counting Segre characteristic types under an unstated convention that eigenvalue values are unlabeled placeholders, then the numerical sequence is correct: the values 14, 27, 58 for n=4,5,6 match A001970, and the n=4 enumeration is accurate under that convention. Theorem 1 is a standard and correct fact, and the paper properly credits the rank-pattern-to-Jordan-block proof. The main value of the note is expository: it connects the Segre-characteristic count to partitions of partitions and provides a small visualization tool. However, the central counting theorem as stated is false, and its proof is circular because P(n) was defined earlier as exactly the count being asserted. The paper therefore does not establish its advertised claim in its current form.

major comments (3)

[Theorem 2] The statement 'There are P(n) Segre Characteristic equivalent matrices of dimension n' is false as written. For every n and every Segre characteristic, there are infinitely many matrices realizing it; for example, the family {[λ] : λ∈C} of 1×1 matrices is infinite, all having Segre characteristic [(1)], while P(1)=1. A finite count is obtained only if a Segre characteristic is treated as an unlabeled multiset of integer partitions, ignoring the actual complex values of the eigenvalues. This convention is not stated in the abstract, in Section 1, or in Theorem 2, and it must be made explicit before the theorem can be true.
[Theorem 2, proof] The proof says the count is 'by definition the partitions of partitions of n,' but P(n) was introduced in Section 1 as exactly that count. The theorem therefore restates the definition rather than deriving the count from the Jordan block structure. A proof needs an explicit bijection between (i) equivalence classes of Jordan block-size lists under grouping by unlabeled eigenvalues and (ii) the objects counted by the generating function ∏_{k≥1}(1−x^k)^{-p(k)}. The n=4 example illustrates the intended correspondence but does not prove it for general n. In addition, the phrase 'unique subsets of the elements in the original partition' is not the correct operation: distributing eigenvalues across blocks corresponds to forming multipartitions or set partitions of the block sizes, not to taking subsets.
[Section 1, definitions and notation] The notation does not cleanly separate an integer partition of n (giving Jordan block sizes for a single eigenvalue) from a Segre characteristic (a multiset of such partitions, one per eigenvalue). The initial example writes [(2,1), 3, 1, (2,1)] with bare integers for singleton blocks, while the enumeration of n=4 writes [3,1], [2,2], etc. for block-size partitions. Without a formal definition of a Segre characteristic as an unlabeled list of integer partitions, Theorem 2's counting claim remains ambiguous even after the eigenvalue-value convention is added.

minor comments (5)

[Title and abstract] There are several typographical errors, including 'EQUIV ALENCE' in the title and 'Partition F unction' and 'F errers Diagrams' in the text.
[Notation, Section 1] The symbols p(n) and P(n) are visually similar and easy to confuse; the paper should consider a more distinct notation, such as π(n) for the partition-of-partitions count.
[Appendix A, Mathematica code] The color list c contains only seven entries, so the visualization routine will not color more than seven distinct eigenvalue labels; for larger n the function either repeats colors or fails.
[Appendix A, Mathematica code] The function M is defined twice with different parameter patterns (once with arguments a, ev and once with a single argument s); this overloading is likely to confuse readers and should be disambiguated.
[Example for n=4] The phrase 'we want to find the unique partitions of the sets of block dimensions above' is informal; the intended operation is the enumeration of multipartitions of the parts of each integer partition of n.

Circularity Check

2 steps flagged · score 8.0 of 10

Theorem 2's count is definitional: the proof states the result is 'by definition the partitions of partitions of n,' and the finite count requires an unstated convention that eigenvalues are unlabeled placeholders.

self definitional [Section 2, Theorem 2 (proof); Section 1, definition of P(n)]
"The number of possible constituent Jordan blocks that an n × n matrix can be comprised of is the integer partitions of n. For each partition, we must count the number of ways we can distribute eigenvalues across the blocks. This is the number of unique subsets of the elements in the original partition. This is by definition the partitions of partitions of n, denoted P(n)."

P(n) was already defined in Section 1 as the number of partitions of partitions, and its generating function is imported from the literature. The proof's 'distribute eigenvalues across the blocks' and 'unique subsets of the elements in the original partition' is exactly the operation that forms a partition of a partition: group the parts of an integer partition into unlabeled blocks. The theorem then concludes 'This is by definition the partitions of partitions of n,' explicitly conceding that the asserted equality holds by construction. No independent enumeration of Jordan-form groupings is derived from the Jordan block structure, rank patterns, or normal-form theory; the worked example simply lists the partitions-of-partitions cases.
other [Abstract and Section 2, Theorem 2 (statement); Section 1, definition of Segre Characteristic]
"Given only the dimension, n, of a square matrix A ∈ M(n,C), how many Segre Characteristic equivalent matrices are there? ... There are P(n) Segre Characteristic equivalent matrices of dimension n."

The theorem promises a finite count 'given only the dimension,' but without a stated convention that eigenvalue values are ignored, the count is not finite: for n=1, the family {[λ] : λ ∈ C} is infinite and every member has Segre characteristic [(1)], so P(1)=1 would be false. The finite count only exists because a Segre Characteristic is defined as an unlabeled grouping of Jordan block sizes by eigenvalue, and the proof's 'distribute eigenvalues across the blocks' treats eigenvalues as indistinguishable placeholders. This unstated convention makes the number of Segre-equivalence classes equal to the number of partitions of partitions by construction.

full rationale

The only substantial derivation in the paper is Theorem 1, which converts a rank pattern into a Segre characteristic via nullity growth and conjugate partitions; that step is standard, is supported by an external textbook, and is not circular. The central counting claim, Theorem 2, is different. P(n) is introduced in Section 1 as the partition-of-partition counting function, with its generating function taken from the literature, and Theorem 2's proof ends with 'This is by definition the partitions of partitions of n, denoted P(n).' The operation described in the proof, distributing eigenvalues across the Jordan blocks and taking unique subsets of the block sizes, is exactly the operation used to define partitions of partitions; no independent enumeration of Jordan-form types is supplied, and the worked examples simply list the partitions of partitions. The finite count also depends on an unstated convention that eigenvalue values are ignored: taken literally, the theorem is false because infinitely many matrices share each Segre characteristic (e.g., scalar matrices λI). Under the intended convention, the theorem is a definitional restatement rather than a derived result. There is no load-bearing self-citation; the circularity is that the central count reduces by construction to the definition of P(n).

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard linear algebra (Jordan form), a classical combinatorial generating function, and an implicit convention that eigenvalue labels are ignored. There are no fitted parameters. The main burden is the unstated relabeling assumption and the fact that the theorem's proof reduces to the definition of P(n).

assumptions (3)

standard math Jordan Normal Form Theorem: every complex square matrix is similar to a unique Jordan form up to block permutation.
Invoked in the abstract and used to frame Segre characteristics as block-size lists; cited to Axler p. 324.
standard math The generating function for the partition-of-partitions count P(n) is ∏_{k≥1}(1−x^k)^{−p(k)}.
Stated in Section 1 without proof; P(n) is defined via this generating function and then used as the answer in Theorem 2.
domain assumption Eigenvalues are treated as unlabeled placeholders, so a Segre characteristic is an unordered multiset of integer partitions, not a matrix-specific object.
Necessary for the finite count P(n); without it, infinitely many matrices share each pattern. The paper never states this convention explicitly, but it is used in the example enumeration and in Theorem 2's proof.

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Cite this review

Pith. "Pith review of Segre Characteristic Equivalence." pith.science (2026). https://pith.science/paper/YZJQJU5X

@misc{pith2026250612065,
  author       = {Pith},
  title        = {Pith review of: Segre Characteristic Equivalence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YZJQJU5X}},
  note         = {Machine review of arXiv:2506.12065}
}

abstract

Given only the dimension, $n$, of a square matrix $A \in M(n,\mathbb{C})$, how many Segre Characteristic equivalent matrices are there? Jordan Normal Form Theorem states that any linear operator over $\mathbb{C}$ is similar to a matrix in Jordan Normal Form. As such, this is a question of counting the number of possible Jordan Normal Forms for a given dimension. So, equivalently, how many Jordan Normal Forms can an $n\times n$ matrix possibly have?

Figures

Figures reproduced from arXiv: 2506.12065 by the authors.

**Figure 2.** All 27 possible Segre Characteristics for 5 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Andrews, G. E. (1976). The Theory of Partitions , volume 2 of Encyclopedia of Mathematics and its Applications . Cambridge University Press

work page 1976
[2]

Axler, S. (2015). Linear Algebra Done Right . Springer, 3rd edition

work page 2015
[3]

Bromwich, T. (1906). Quadratic Forms and Their Classification by Means of Invariant-Factors . Cambridge University Press

work page 1906
[4]

A., Duncan, W

Frazer, R. A., Duncan, W. J., and Collar, A. R. (1963). Elementary Matrices and Some Applications to Dynamics and Differential Equations . Cambridge University Press. First published in 1938, reprinted in 1963

work page 1963
[5]

I.R.Shafarevich and Remizov, A. (2013). Linear Algebra and Geometry . Springer

work page 2013
[6]

Sylvester, J. (1854). Note on the enumeration of the contacts of lines and surfaces of the second order. Philosophical Magazine , 7:331--334

work page

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Andrews, G. E. (1976). The Theory of Partitions , volume 2 of Encyclopedia of Mathematics and its Applications . Cambridge University Press

work page 1976

[2] [2]

Axler, S. (2015). Linear Algebra Done Right . Springer, 3rd edition

work page 2015

[3] [3]

Bromwich, T. (1906). Quadratic Forms and Their Classification by Means of Invariant-Factors . Cambridge University Press

work page 1906

[4] [4]

A., Duncan, W

Frazer, R. A., Duncan, W. J., and Collar, A. R. (1963). Elementary Matrices and Some Applications to Dynamics and Differential Equations . Cambridge University Press. First published in 1938, reprinted in 1963

work page 1963

[5] [5]

I.R.Shafarevich and Remizov, A. (2013). Linear Algebra and Geometry . Springer

work page 2013

[6] [6]

Sylvester, J. (1854). Note on the enumeration of the contacts of lines and surfaces of the second order. Philosophical Magazine , 7:331--334

work page