The Group Theoretic Roots of Information: permutations, symmetry, and entropy

David J. Galas

REVIEW 3 major objections 4 minor 2 references

The Group Theoretic Roots of Information: permutations, symmetry, and entropy

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Shannon entropy of a partition is the large-N limit of an entropy defined by permutation cycle counts, and every finite group carries its own entropy.

desk verdict Genuinely new combinatorial entropy whose central limit theorem is false as stated; the subgroup generalization is underdefined, but the cycle-polynomial machinery is worth a careful revision. read the letter →

arxiv 1908.09642 v2 pith:LXGCCA3Q submitted 2019-08-23 cs.IT math.IT

David J. Galas This is my paper · ORCID

classification cs.ITmath.IT MSC 94A1705A1520B0560C05

keywords integerentropypermutationcyclesShannonfinitegroupsharmonicnumberscyclepolynomialtranspositionsymmetrybreaking

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

This paper sets out to show that the Shannon entropy of a partition $\{n_i\}$ of $N$ objects is the large-$N$ limit of a purely combinatorial quantity built from permutation cycles: the integer entropy $J_{G_N}$, defined as the expected number of cycles under the group $G_N$ minus the $N$-weighted sum of the expected cycle counts inside each part. For the symmetric group $S_N$, the paper computes $J_{S_N}(\{n_i\})=H_N-\frac{1}{N}\sum_i H_{n_i}$, with $H_n$ the $n$-th harmonic number, and argues that as $N$ and every $n_i$ grow, this tends to $I(\{n_i\})=-\sum_i \frac{n_i}{N}\ln\frac{n_i}{N}$. The reason to care is that Shannon entropy then appears as the symmetric-group endpoint of a family of information functionals, one for every finite group, obtained by restricting the allowed permutations to a subgroup. Each such group carries its own harmonic-analog series from its cycle polynomial, so information becomes tied to symmetry rather than to a list of axioms.

What carries the argument

The workhorse is the cycle polynomial $P_G(x)=\sum_{\pi\in G}x^{c(\pi)}$, where $c(\pi)$ is the number of disjoint cycles in permutation $\pi$. For the symmetric group it is the rising factorial $x^{\overline{N}}=x(x+1)\cdots(x+N-1)$, and the expected cycle number is the logarithmic derivative $P'_G(1)/P_G(1)$; equivalently $\langle C\rangle_G=\sum_k(1-r_k)^{-1}$ with $r_k$ the roots of $P_G$. The integer entropy compares $\langle C\rangle_{G_N}$ with the $N$-weighted average $\frac{1}{N}\sum_i n_i\langle C\rangle_{G_{n_i}}$. The transposition polynomial $Q_G(x)=\sum_{\pi\in G}x^{T(\pi)}$, built from the Cayley relation $T(\pi)=N-c(\pi)$, is the reciprocal (reflected) polynomial of $P_G$, and its roots are the reciprocals of the cycle-polynomial roots.

What would settle it

A direct calculation from Eq. (8) refutes the unqualified limit: take the partition with $k$ parts of size 1 and one part of size $N-k$, with $k$ fixed, and let $N\to\infty$. The claimed limit, Shannon entropy, tends to $0$, while the exact expression $H_N-(kH_1+H_{N-k})/N$ tends to $\ln N+\gamma$, so the difference diverges; this tests the missing uniformity in Theorem 1 without any simulation.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: in the double limit where the group becomes the full symmetric group $S_N$ and then $N$ tends to infinity, the integer entropy $J_{S_N}$ equals Shannon entropy $I(\{n_i\})$. The proof uses the fact that the average cycle count of a uniformly random permutation of $n$ objects is exactly $H_n$, via the signless Stirling numbers; then $J_{S_N}=H_N-\frac{1}{N}\sum_i H_{n_i}$, and replacing each $H_{n_i}$ by $\ln n_i+\gamma$ leaves $\ln N-\frac{1}{N}\sum_i\ln n_i$, which is $I(\{n_i\})$ because $n_i/N=p_i$. For every subgroup $G\subseteq S_N$, the same expected-cycle construction defines an integer entropy $J_G(\{n_i\})$, and the logarithmic derivative of the Cameron-Semeraro cycle polynomial at $x=1$ gives the group's expected cycle number, also expressible as $\sum_k(1-r_k)^{-1}$ over roots. The paper also introduces a reciprocal transposition polynomial whose logarithmic derivative gives the expected minimum number of transpositions needed to undo a permutation.

Load-bearing premise

The load-bearing premise is that every part $n_i$ grows to infinity with $N$ so that $H_{n_i}$ can be replaced by $\ln n_i+\gamma$, and that a subgroup $G$ of $S_N$ induces a well-defined permutation group on every subset of size $n_i$; the paper proves neither uniformly.

Editorial extensions

If this is right

Shannon entropy becomes the $S_N$ case of a wider family: restricting permutations to a subgroup $G$ yields a conditional entropy that quantifies disorder relative to that symmetry, and every finite group has such a functional.
Every finite permutation group produces a harmonic-like number series, its expected cycle count, computed from the logarithmic derivative of its cycle polynomial; the paper illustrates the alternating, cyclic, and dihedral series and shows some converge to the harmonic numbers while others, like the cyclic-group series, follow a divisor-controlled pattern.
The reciprocal transposition polynomial gives the expected minimum transposition count from the same roots, and defines balanced groups with $\langle T\rangle=\langle C\rangle=n/2$; for these, the integer entropy takes the simple form $J_B(\mathrm{uniform})=\frac{N}{2}\frac{m-1}{m}$.
Because cycle structure is constant on conjugacy classes, the expected cycle numbers and the integer entropy depend only on conjugacy-class data of the group.

Reading between the lines

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

A renormalized version of the integer entropy that uses exact harmonic differences rather than replacing $H_{n_i}$ by $\ln n_i+\gamma$ would be well-defined for partitions with bounded part sizes, and comparing it with Shannon entropy across partitions would quantify how quickly the symmetric-group limit is approached.
Because $\langle C\rangle_G=\sum_k(1-r_k)^{-1}$ depends only on the multiset of roots of the cycle polynomial, the integer entropy may depend on how an abstract group is embedded as a permutation group; computing it for two non-conjugate permutation representations of the same group would settle this directly.
The cyclic-group result, where expected cycle counts for prime order converge to 2 while non-prime orders show divisor-driven scatter, suggests a statistical test: for random $n$, the expected cycle count should show more variance than the harmonic series, with variance predictable from the divisor function.

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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

Here's the short version: the paper has a genuinely new object — the integer entropy J_G defined through expected cycle counts over a permutation group — and the cleanly stated cycle-polynomial results are mostly correct. But the central limit theorem is not true as stated, and the advertised extension to all finite groups is not actually defined. Don't desk-reject it, but send it back for major surgery.

What's new: the functional J_G({n_i}) = _{G_N} - (1/N) Σ n_i _{G_{n_i}} appears to be new, and using the Cameron-Semeraro cycle polynomial to compute it is a nice idea. Theorem 3 (logarithmic derivative gives expected cycle count) and Theorems 4-6 (root formulas for cycle and transposition averages) are correct, standard manipulations. The harmonic-number-as-expected-cycles fact is classical, but the paper's repackaging through cycle polynomials does give a useful language.

The soft spots are real. Theorem 1, the paper's centerpiece, claims a limit to Shannon entropy, with 'uniform' convergence in the abstract. The proof replaces every H_{n_i} by ln n_i + γ and lets the γ terms cancel. That cancellation is only legitimate when every part n_i grows with N. The paper states no such condition. Take N=2M with M parts equal to 1 and one part equal to M: direct calculation gives J = H_{2M} - 1/2 - (1/2)H_M, which tends to I + (γ−1)/2, not I. So Eq. 4 is false as written and 'uniform' is simply wrong. The fix is easy — require all parts to grow — but it has to be stated, and the statement of Theorem 1 has to match.

The deeper issue is that for a general subgroup G_N of S_N, the expression uses G_{n_i}, a permutation group on each subset of size n_i. A subgroup of S_N does not canonically restrict to a permutation group on an arbitrary subset; the paper never defines G_{n_i}. So the promised 'integer entropy for every finite group' is not well-defined without extra structure. This is load-bearing for the generalization, though the symmetric-group case is fine once the limit condition is repaired.

One smaller thing: Appendix B's 'balanced group' example deletes two transpositions from S_4 and claims the result is still a group. It isn't — 22 elements can't be a subgroup of S_4, and the set isn't closed. That example needs to be fixed or dropped.

The citation pattern is reasonable, and there's no data to worry about; the content is the math. The paper is worth a serious referee. The errors are specific and fixable, and the core combinatorial object may be worth keeping. I wouldn't cite it in its current form, but I'd read a careful revision.

Referee Report

3 major / 4 minor

Summary. The paper introduces an 'integer entropy' functional J_G for a permutation group G_N on N objects and a partition {n_i} of N, defined through expected numbers of cycles in the group and in group actions on the parts. For the symmetric group it reduces to J_{S_N}=H_N-(1/N)Σ n_i H_{n_i}. The central claim (Theorem 1, Eq. (4)) is that the N→∞ limit of J_{S_N} equals the Shannon entropy of the probability vector {n_i/N}, and the paper states this convergence is uniform. The paper then develops the Cameron–Semeraro cycle polynomial and a reciprocal 'transposition polynomial,' giving expressions for expected cycle and transposition counts in terms of polynomial roots, and illustrates the framework on alternating, cyclic, dihedral, and 'balanced' groups. An appendix also discusses broken symmetries and order parameters for the dihedral group.

Significance. If the central limit theorem were correct, the paper would establish a new combinatorial bridge between finite permutation groups and Shannon information, with a concrete number series for every finite group. The cycle-polynomial results (Theorems 3–6) are standard and are derived cleanly; the reciprocality between the cycle and transposition polynomials and the root-sum formulas are correct and potentially useful. The appendix computations for A_n, C_n, D_n, and balanced groups are explicit and make the framework concrete. However, as explained in the major comments, the principal limit theorem is false as stated, and the definition of the integer entropy for arbitrary subgroups is incomplete; these issues must be repaired before the paper's central claims can be accepted.

major comments (3)

[§II, Theorem 1 (Eq. (4)) and proof (Eqs. (8)–(9))] The proof of Theorem 1 replaces every harmonic number H_{n_i} by ln n_i + gamma, which is legitimate only when every n_i tends to infinity. As stated, the theorem and the abstract claim uniform convergence without this condition, and the claim is false without it. For example, take N=2M and the partition consisting of M parts of size 1 together with one part of size M; then Eq. (8) gives J_{S_N}=H_{2M}-1/2-(1/2)H_M, whose limit is (1/2)ln N + (1/2)ln 2 + (gamma-1)/2, whereas the Shannon entropy of the same probability vector is (1/2)ln N + (1/2)ln 2. The difference is the nonzero constant (gamma-1)/2. A correct statement needs an explicit growth condition, for example that the total probability mass carried by parts that remain bounded tends to zero, or that all n_i tend to infinity.
[§III, Eq. (14a) and Eq. (14b)] The formula for the integer entropy in Eq. (14a), J_G({n_i}) = <C(N)>_G - ∑ n_i <C(n_i)>_{G_i}, and its polynomial form in Eq. (14b), are missing the factor 1/N that appears in the definition Eq. (3) and in the symmetric-group expression Eq. (8). This is not a notational slip: with the factor omitted, J_{S_N} would become H_N - ∑ n_i H_{n_i}, which diverges even in simple partitions, and the appendix calculations based on Eq. (14) would be inconsistent with the main definition. The factor must be restored consistently throughout Section III and the appendix.
[§II, Eq. (3) and §III, Eq. (14)] The definition of J_G({n_i}) uses the quantities <C>_{G_{n_i}}, the expected cycle counts of the group acting on each subset of size n_i. For the symmetric group these are simply H_{n_i}, but for an arbitrary subgroup G⊆S_N there is no canonical permutation group induced on an arbitrary subset of size n_i, and the paper never defines G_{n_i}. Consequently the 'integer entropy of a finite group' is not well-defined unless one specifies, for every part of the partition, a compatible permutation representation of G on that part. This is a load-bearing gap for the claim that every finite group has a corresponding information functional; it affects not only Eq. (3) but also the examples in Appendix A.

minor comments (4)

[§V] The name 'Cayley' is misspelled as 'Caley' in the sentence 'a result from Caley in 1849', although the reference list correctly attributes Cayley.
[§II, Eq. (1) and Eq. (9)] Shannon entropy is denoted I in Eq. (1) but S in Eq. (9); please use a single notation throughout.
[§III, proof of Theorem 2] The displayed equation (11) is garbled; the summation over group elements and the Kronecker delta are not printed clearly, making the proof harder to follow than necessary.
[Appendix A.2] In Table A2 and the surrounding text, the phrase 'where the d are the number of divisors of n' is awkward; it should read 'where d ranges over the divisors of n'.

Circularity Check

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No significant circularity: Theorem 1 follows from a direct asymptotic calculation on harmonic numbers, and the integer entropy is not defined in terms of the Shannon entropy.

full rationale

The central claim (Eq. 4) is derived from Eq. 8, J_{S_N}=H_N-(1/N)Σ n_i H_{n_i}, using only the standard harmonic-number asymptotic H_n=\ln n+γ+o(1). The integer entropy in Eq. 3 is defined via expected permutation-cycle counts, not via the Shannon entropy, and the combinatorial interpretation of H_n as an average cycle count supplies independent grounding; no fitted parameter is renamed as a prediction and no load-bearing self-citation supports the limit. The author's earlier work appears only as background on multivariate information measures. The proof is self-contained apart from the quoted standard identity for signless Stirling numbers. The theorem's uniformity claim is not proven and the limit can fail when a nonvanishing fraction of partition mass lies in bounded parts, and the notation G_{n_i} for arbitrary subgroups is never defined; these are correctness gaps and not instances of circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The central claim rests on standard combinatorial facts (harmonic numbers, cycle polynomials) plus two ad hoc assumptions: that the integer entropy is a meaningful information measure, and that a permutation group on N objects induces well-defined permutation groups on the parts of a partition. The latter is not defined. No free parameters are fitted.

assumptions (7)

standard math The expected number of cycles of a uniformly random permutation in S_n is the nth harmonic number H_n.
Invoked in Equation 7 and the proof of Theorem 1; a classical combinatorial fact.
standard math lim_{n->infinity} (H_n - ln n) = gamma, the Euler-Mascheroni constant.
Used to replace harmonic numbers by logarithms in Equation 9 and Theorem 1.
standard math Every finite group is isomorphic to a subgroup of some symmetric group (Cayley's theorem).
Used in the Introduction to justify treating all finite groups via permutation subgroups.
standard math The cycle polynomial and its logarithmic derivative give expected cycle counts for a permutation group.
Theorem 3, following the Cameron-Semeraro cycle polynomial definition.
ad hoc to paper The integer entropy defined in Equation 3 is a valid information measure.
The paper postulates this measure; its physical or information-theoretic meaning is not independently justified beyond its limit to Shannon entropy.
domain assumption In the limit for the symmetric group, every part n_i of the partition grows without bound (implicit).
Needed for the pointwise limit in Theorem 1; not stated, and fails for partitions with bounded n_i, contradicting the claimed uniform convergence.
ad hoc to paper For a subgroup G and a partition {n_i}, there is a well-defined group G_{n_i} acting on each subset, with expected cycle counts <C>_{G_{n_i}}.
Required by the definition of J_G in Equations 3 and 14a; the paper does not define how a subgroup of S_N induces such groups on arbitrary subsets.

invented entities (3)

Integer entropy J_G
purpose: Generalized entropy functional for permutation groups
Defined in Equation 3; no independent empirical handle; its main property is its Shannon limit.
Transposition polynomial Q_G(x)
purpose: Tool to compute expected transposition counts and bound coefficients
Defined in Section V as reciprocal to the cycle polynomial; a mathematical construction without external evidence.
Balanced groups
purpose: Class of groups where average cycle count equals average transposition count
Defined in Appendix B; introduced as an example, not observed in nature or prior literature.

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Cite this review

Pith. "Pith review of The Group Theoretic Roots of Information: permutations, symmetry, and entropy." pith.science (2026). https://pith.science/paper/LXGCCA3Q

@misc{pith2026190809642,
  author       = {Pith},
  title        = {Pith review of: The Group Theoretic Roots of Information: permutations, symmetry, and entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXGCCA3Q}},
  note         = {Machine review of arXiv:1908.09642}
}

read the original abstract

We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial measure of information and disorder is proposed, in terms of integers and discrete functions, that we call the integer entropy. The Shannon measure of information is the limiting case of a richer, more general conceptual structure that reveals relations among finite groups, information, and symmetries. It is shown that the integer entropy converges uniformly to the Shannon entropy when the group includes all permutations, the Symmetric group, and the number of objects increases without bound. The harmonic numbers have a well-known combinatorial meaning as the expected number of disjoint, non-empty cycles in permutations of n objects, and since integer entropy is defined in terms of the expected value of the number of cycles over the set of permutations, it also has a clear combinatorial meaning. Since all finite groups are isomorphic to subgroups of the Symmetric group, every finite group has a corresponding information functional, analogous to the Shannon entropy and a number series analogous to the harmonic numbers. The Cameron-Semeraro cycle polynomial is used to analyze the integer entropy for finite groups, and to characterize the series analogous to the Harmonic numbers. We introduce and use a reciprocal polynomial, the transposition polynomial that provides an additional tool and new insights. Broken symmetries and conserved quantities are linked through the cycle and transposition properties of the groups, and can be used to generalize the analysis of stochastic processes.

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Works this paper leans on

2 extracted references · 1 canonical work pages

[9]

Dynamical foundations of nonextensive statistical mechanics

Beck, C., “Dynamical foundations of nonextensive statistical mechanics.” Phys Rev Lett, 87(18), 180601 (2001). 10. Beck, C., Cohen, E.G.D., “Superstatistics.” Physica A, 322:267–275 (2003). 11. Galas D.J., Sakhanenko N.A., Skupin A, Ignac T., “Describing the Complexity of Systems: Multivariable "Set Complexity" and the Information Basis of Systems Biology...

arXiv 2001
[26]

The cycle polynomial of a permutation group

Postnikov, A., ”Permutohedra, associahedra, and beyond.” Int Math Res Not, 2009(6): 1026-1106 (2009). 27. Roman, S., The Umbral Calculus, (p-63) Academic Press, NY, (1984). 28. Cameron, P.J. and Semeraro, J., ”The cycle polynomial of a permutation group.” arXiv: 1701.06954 (2017). 29. Cameron, P.J., Jackson, B. and Rudd, J.D., “Orbit-counting polynomials ...

work page Pith review arXiv 2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Dynamical foundations of nonextensive statistical mechanics

Beck, C., “Dynamical foundations of nonextensive statistical mechanics.” Phys Rev Lett, 87(18), 180601 (2001). 10. Beck, C., Cohen, E.G.D., “Superstatistics.” Physica A, 322:267–275 (2003). 11. Galas D.J., Sakhanenko N.A., Skupin A, Ignac T., “Describing the Complexity of Systems: Multivariable "Set Complexity" and the Information Basis of Systems Biology...

arXiv 2001

[2] [26]

The cycle polynomial of a permutation group

Postnikov, A., ”Permutohedra, associahedra, and beyond.” Int Math Res Not, 2009(6): 1026-1106 (2009). 27. Roman, S., The Umbral Calculus, (p-63) Academic Press, NY, (1984). 28. Cameron, P.J. and Semeraro, J., ”The cycle polynomial of a permutation group.” arXiv: 1701.06954 (2017). 29. Cameron, P.J., Jackson, B. and Rudd, J.D., “Orbit-counting polynomials ...

work page Pith review arXiv 2009