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Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology

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Pith's one-line read Cycle-decorated ribbon bar complexes canonically split into classical ribbon complexes, yielding complete bigraded equivariant homology with ribbon-positive representations whenever the composition has at most one odd part.

desk verdict A genuinely new homology computation for cycle-decorated ribbon bar complexes, with a load-bearing factorization lemma that needs a fuller proof before I'd trust Theorem 3.24 completely. read the letter →

arxiv 2608.07599 v1 pith:I7EB5O2N submitted 2026-08-06 math.CO math.AT

Pyuyi Chufeng Huang This is my paper · ORCID
classification math.COmath.AT MSC 05E1605A1505E0506A0713A3516T0552B2055U10
keywords ribbonbarcomplexesequivarianthomologyrootedpermutationsSchurfunctionsnoncommutativesymmetricorderpolynomialsoffencesenrichedchainpolytopesalgebraicdiscreteMorsetheory
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 is trying to establish that a large family of equivariant chain complexes—ordered-set-partition bars decorated by one ordinary and one rooted permutation—decomposes canonically into the classical ribbon bar complexes whose homology was already known. The decomposition is driven by a single operation, root-tail concatenation of rooted permutations, whose simultaneous unique factorization along cuts is proved for every composition with at most one odd part. If correct, the main theorem gives the complete bigraded $S_n$-equivariant homology of these complexes for that whole family: every homology representation is ribbon-positive, and the multiplicity of each ribbon is a positive weight enumerator counting decorations with a prescribed exact cut set. The same machinery connects the character to alternating-fence order polynomials, enriched chain polytopes, and the exact Hilbert–Kunz formula for quadrics.

What carries the argument

The load-bearing object is the rooted-cycle decoration algebra. A decoration of a block of size $m$ is a pair $(\sigma,\tau)$, where $\sigma$ is an ordinary permutation of $r_m=\lceil m/2\rceil$ letters and $\tau$ is a rooted permutation of $s_m+1$ letters with one distinguished root $*$; the bidegree records cycle counts: $\deg_q=r_m-c(\sigma)$ and $\deg_t=c(\sigma)+c(\tau)-1$. Adjacent blocks multiply by direct sum of ordinary permutations and root-tail concatenation of rooted permutations, and the product is declared zero when both block sizes are odd. The mechanism that carries the argument is Lemma 3.23, the simultaneous unique-factorization theorem: for a composition with at most one odd part, a total decoration factors uniquely along any set of cuts, and it factors along precisely those cuts that its factorization-cut set contains. This turns the decorated complex into a direct sum of classical rank-selected Boolean ribbon complexes, whose homology was already known.

What would settle it

Enumerate all total decorations $\theta\in\mathcal D_5$ for $\alpha=(3,2)$, compute $\mu_\beta$ for every coarsening $\beta$, and check that no two distinct local-factor tuples give the same $\theta$ and that each $\theta$ with $D(\beta)\subseteq F_\alpha(\theta)$ actually lies in the image; a single duplicated product, or a decoration that factors despite a missing cut, would refute Lemma 3.23 and Theorem 3.24.

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Extended reading notes

Core claim

The central claim is Theorem 3.24: for every composition $\alpha\models n$ with at most one odd part, total decoration induces a canonical, bidegree-preserving, $S_n$-equivariant chain isomorphism $C^{t,q}_\bullet(\alpha)\cong\bigoplus_{\theta\in\mathcal D_n}C_\bullet(\gamma_\alpha(\theta))$. Consequently $\operatorname{ch}_{t,q}H_k(C^{t,q}_\bullet(\alpha))$ equals the sum over decorations $\theta$ whose exact factorization-cut set has size $k-1$ of $t^{\deg_t\theta}q^{\deg_q\theta}r_{\gamma_\alpha(\theta)}$, where $r_\gamma$ is the ribbon Schur function. In words, every bigraded homology representation is ribbon-positive and every multiplicity is the weighted count of decorations factorable along precisely that cut set. The author derives this from Lemma 3.23, which says the iterated decoration product is injective on any compatible family of block sizes and has image exactly the decorations whose factorization cuts contain the coarsening's cut set.

Load-bearing premise

The load-bearing premise is Lemma 3.23: for any composition with at most one odd part, every total decoration factors uniquely along exactly those cuts that its factorization-cut set contains—if that simultaneous unique factorization failed for any composition, the canonical splitting and all homology formulas would collapse.

Editorial extensions

If this is right

  • For every composition with at most one odd part, the bigraded equivariant homology is completely known: each homology module is ribbon-positive, hence Schur-positive, with explicit weighted multiplicities.
  • In top bar degree, the homology has Frobenius characteristic $(\prod_i Q_{\alpha_i}(t,q))r_\alpha$, and for staircase compositions this specializes to $t^{\lceil n/2\rceil}(1+t)^{\lfloor n/2\rfloor}r_{\delta_n}$, lying in $q$-degree zero.
  • On staircase ribbons, setting $q=-1$ recovers the order polynomial of the alternating fence, so one two-parameter ribbon character contains both parity families of fence order polynomials.
  • The extreme graded homology strata of the staircase complex are explicitly bijective with the extreme fibers of the zigzag-record statistic, while a Betti-number obstruction rules out any Morse compression along the given differential to one cell per permutation.
  • The orthant-gluing resolution gives an extended Fibonacci recurrence for enriched chain polytopes and lifts both Ehrhart terms in the exact Hilbert–Kunz formula for quadrics to staircase ribbon characters.

Reading between the lines

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

  • The named mechanism—cut-factorizing decoration systems—is likely to work for other monoids whose multiplication has simultaneous unique factorization, so one could build explicit homology decompositions by swapping the Boolean ribbon summands for rank-selected geometric-lattice complexes.
  • The proof of sign coherence on integral rays $t=m u$ suggests that $n!Z_{\delta_n}(t,-u)$ is the generating function of a single permutation statistic refining zigzag records by a cycle defect; finding such a statistic would settle the full conjecture beyond $n=22$.
  • The rigidity theorem implies that any genuinely smaller chain model of the even-block bar must change the coproduct or use transferred homotopy-bialgebra operations, so the paper's normalization is the boundary of ordinary strict Hopf-compatible reduction.
  • The chain-level splitting may yield stability phenomena as $n$ grows: the decorated homology modules for staircase compositions might stabilize in a grading-dependent way, parallel to known stability in rank-selected homology.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces bigraded S_n-chain complexes whose ordered set partitions carry decorations by an ordinary permutation and a rooted permutation. The Hilbert–Euler characteristic of these complexes is n! Z_α(t,q) for a two-parameter character of noncommutative symmetric functions. The main chain-level claim is that, when the composition α has at most one odd part, simultaneous unique factorization of total decorations yields a canonical bidegree-preserving, S_n-equivariant splitting into classical ribbon bar complexes, C^{t,q}_•(α) ≅ ⊕_{θ∈D_n} C_•(γ_α(θ)). From this the paper derives a complete ribbon-positive formula for the bigraded equivariant homology, with multiplicities counting decorations with prescribed exact factorization-cut sets. Secondary results include a staircase specialization to alternating fence order polynomials, a sign-coherence theorem on integral rays, an extreme-record-fiber identification, a Morse obstruction, an orthant-gluing resolution of enriched chain polytopes and its Hilbert–Kunz reformulation, a commutative Hopf bar realization, and a rigidity theorem for Hopf-compatible Morse normalizations.

Significance. If the central decomposition is correct, the paper gives a rather complete equivariant homology calculation for a new family of decorated bar complexes and exhibits explicit ribbon-positive Frobenius characteristics. The main construction is elegant, with precise theorem statements and many standard proofs; the exact character formulas and the explicit fiber decomposition are strong points. The paper also makes several concrete connections to fence order polynomials, record statistics, enriched chain polytopes, and Hilbert–Kunz multiplicity, and it provides exact-arithmetic scripts for finite verification. However, two issues currently prevent full confidence: the proof of the load-bearing unique-factorization lemma is too terse, and one technical lemma in the integral-ray sign-coherence argument is based on a demonstrably false recurrence. These are localized but must be repaired before the paper's claims can be relied upon.

major comments (2)
  1. [§3.10, Lemma 3.23 and Theorem 3.24] Lemma 3.23 is the load-bearing step for the canonical splitting (62), but its proof is a single paragraph. Please replace it by a complete argument that spells out the factorization algorithm: how the ordinary factors are recovered by restriction and standardization at the boundaries r_{c_j}, how the rooted factors are recovered from the root-tail subwords together with the confined nonroot cycles, and why the simultaneous conditions for nested cuts are exactly equivalent to membership in the image of μ_β. The edge cases need explicit treatment: empty root-tail prefixes or suffixes, cuts with s_c = 0 or s_c = s_n, odd parts of size 1, and nonroot cycles in an interval whose root-tail subword is empty. As written, the paragraph asserts simultaneous compatibility rather than proving it, and Theorem 3.24 inherits this gap.
  2. [§4.3, Lemma 4.6, Eq. (99)] The contiguous relation (99) is false. At z = 0 with m = 2 and c = 1/2, the left-hand side (m+c−1)G_m equals 3/2, while the right-hand side (2m−2+c)G_{m−1} − (m−1)(1−z)G_{m−2} equals −1/2; the same mismatch occurs in the coefficient of z for arbitrary b. Since the definitions of T_m, S_m, U_m and the induction in Lemma 4.6 all rely on (99), the proof of Lemma 4.6 and hence the proof of Proposition 4.7 (cycle sign coherence on integral rays) are not currently supported. Please correct the recurrence or supply a different proof of the positivity assertion.
minor comments (4)
  1. [Definition 3.19 and Eq. (129)] The same symbol C^{t,q}_•(α) is used for the ordered-product complex in Definition 3.19 and for the averaged-product complex in (129). Please use distinct notation for the two lifts, or explicitly announce that the symbol is being reused in Section 6.
  2. [Proposition 3.7] The phrase 'after the degree shift C_k ↔ C̃_{k−2}' is ambiguous; writing C_k ≅ C̃_{k−2} would state the intended isomorphism more clearly.
  3. [Declarations, Data and code availability] The five verification scripts are described but not included in the text; since the declarations state that they are not used as proofs, this is acceptable, but please make the ancillary files available and include the main numerical outputs so the stated finite checks can be reproduced from the paper alone.
  4. [Theorem 5.1] In the proof, the degree-zero homology 'basis' for a zero coordinate of y is not literally a single basis vector but the class of e_+ + e_−; a short clarifying sentence would prevent confusion in the identification with signed lattice points.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: only self-citation [14] is non-load-bearing.

full rationale

The paper's central derivation is self-contained. The character Z_alpha(t,q) is defined in Definition 2.2 by a prescribed formula on the complete generators H_m, and the decorated complex is constructed later in Definition 3.19. The fact that the shifted Hilbert-Euler polynomial of the complex equals n!Z_alpha(t,q) is a matching by construction, but the paper does not present that matching as the main prediction. The main theorem, Theorem 3.24, is a genuine homology computation: it depends on Lemma 3.23, which proves simultaneous unique factorization of rooted-cycle decorations by an internal interval and root-tail argument, and on the classical rank-selected Boolean homology theorem, cited externally as [31, 32, 37]. The formula ch_{t,q}H_k(C^{t,q}_•(α)) = sum over θ with |F_alpha(θ)|=k-1 of t^{deg_t θ}q^{deg_q θ} r_{γ_alpha(θ)} is therefore derived, not assumed. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is imported to forbid alternatives. The only self-citation is [14], used for the zzrec record formula in connection with the extreme fibers of the staircase homology. That citation is explicitly paired with the external reflected block formula of [17], and it is not needed for the total-decoration splitting or for the ribbon-positivity result; Theorem 3.29 uses it only for a bijective interpretation of two extreme graded strata, and Theorem 3.30's Morse obstruction is independent of [14]. The enriched chain polytope and Hilbert-Kunz applications use external results [25] and [33]. The terseness of the Lemma 3.23 proof is a legitimate completeness concern but not a circularity: no reduction of the theorem to its own conclusions is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central results rest on standard algebraic combinatorics and topology (rank-selected Boolean homology, Kreweras determinants, discrete Morse theory, species Hopf monoids), plus two external results used as benchmarks ([14] record model and [25] Hilbert-Kunz formula). No free parameters are fitted; the new combinatorial objects are explicitly defined and checked on small cases.

assumptions (6)
  • standard math Rank-selected Boolean poset homology: the homology of B_n(D) is concentrated in the top rank and has Frobenius characteristic r_α.
    Invoked in Theorem 3.8 and used for all homology computations in the paper.
  • standard math Kreweras determinant formula for order polynomials of skew shapes.
    Used in Theorem 2.8 to prove Z_{δ_n}(t,-1)=Ω(P_n;t).
  • domain assumption The record-statistic formula Σ t^{zzrec(π)} = n! Ω(P_n;t) from Huang [14].
    External result used only to interpret extreme homology fibers in Theorem 3.29; the bijections are proved directly.
  • domain assumption Exact Hilbert-Kunz formula for quadrics from Pak-Shapiro-Smirnov-Yoshida [25].
    Used in Corollary 5.5 to rewrite both Ehrhart terms; the paper proves no new Hilbert-Kunz inequality.
  • standard math Algebraic discrete Morse theory and the standard shuffle-deconcatenation bar construction for connected commutative monoids.
    Underpins the Hopf-compatible normalization and rigidity results in Section 7.
  • standard math Stanley's transfer theorem for chain polytopes: |mC_P ∩ Z^I| = Ω(P;m+1).
    Used in Theorem 5.1 to pass from orthant sums to order polynomials.
invented entities (2)
  • Rooted permutations R_s (permutations of s+1 letters with distinguished root *) independent evidence
    purpose: Serve as the second decoration component; their cycle count raises the t-degree and their root-tail concatenation gives the non-odd factorization.
    The definition is explicit, the enumerator t(t+1)...(t+s) is independently computable, and small cases are checked in the scripts.
  • Cycle-decorated ribbon bar complexes C^{t,q}_•(α) with ordered and averaged products independent evidence
    purpose: Realize the character Z_α at chain level and support the homology decomposition.
    The complexes are explicitly defined, their Euler characteristic is computed by design, and the homology decomposition is verified for small cases by scripts.

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Pith. "Pith review of Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology." pith.science (2026). https://pith.science/paper/I7EB5O2N

@misc{pith2026260807599,
  author       = {Pith},
  title        = {Pith review of: Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7EB5O2N}},
  note         = {Machine review of arXiv:2608.07599}
}
abstract

We introduce bigraded $\mathfrak S_n$-complexes whose ordered-set-partition bars are decorated by an ordinary and a rooted permutation. Their Hilbert--Euler characteristic is $n!Z_\alpha(t,q)$, obtained from a two-parameter character of noncommutative symmetric functions. When the composition $\alpha$ has at most one odd part, simultaneous unique factorization of total decorations gives a canonical splitting \[ C_\bullet^{t,q}(\alpha)\cong\bigoplus_{\theta\in\mathcal D_n}C_\bullet(\gamma_\alpha(\theta)) \] into classical ribbon complexes. We thereby determine every bigraded homology representation: its Frobenius characteristic is ribbon-positive, with multiplicities counting decorations with prescribed exact cut sets. For the staircase compositions $\delta_n$, the specialization $Z_{\delta_n}(t,-1)$ is the order polynomial of the alternating fence. Its conjectural cycle sign pattern holds on every integral ray $t=m\,u$. Its extreme homology strata realize the extreme fibers of a greedy-record model, whereas a Betti-number obstruction rules out direct Morse compression to one cell per permutation. We also refine the orthant decomposition of enriched chain polytopes by an explicit resolution. For alternating fences it gives an extended Fibonacci recurrence and a Grothendieck lift of both Ehrhart terms in the exact Hilbert--Kunz formula for quadrics. Finally, an averaged decoration product places the complexes for all compositions in a differential graded Hopf bar. The classical normalization matching is strictly Hopf-compatible, while a rigidity theorem rules out any further nonidentity normalized contraction that preserves the same deconcatenation on the reduced even-block model.

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Works this paper leans on

40 extracted references · 23 canonical work pages

  1. [1]

    Aguiar and S

    M. Aguiar and S. Mahajan.Monoidal Functors, Species and Hopf Algebras, volume 29 ofCRM Monograph Series. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/crmm/029

    doi:10.1090/crmm/029 2010

  2. [2]

    Aguiar and S

    M. Aguiar and S. Mahajan. Hopf monoids in the category of species. InHopf Algebras and Tensor Categories, volume 585 ofContemporary Mathematics, pages 17–124. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/conm/585/11665. CYCLE-DECORATED RIBBON BAR COMPLEXES 35

    work page doi:10.1090/conm/585/11665 2013

  3. [3]

    Aguiar and F

    M. Aguiar and F. Sottile. Structure of the Malvenuto–Reutenauer Hopf algebra of permutations. Advances in Mathematics, 191(2):225–275, 2005. doi: 10.1016/j.aim.2004.03.007

    work page doi:10.1016/j.aim.2004.03.007 2005

  4. [4]

    Aguiar, N

    M. Aguiar, N. Bergeron, and F. Sottile. Combinatorial Hopf algebras and generalized Dehn–Sommerville relations.Compositio Mathematica, 142(1):1–30, 2006. doi: 10.1112/ S0010437X0500165X

    work page 2006

  5. [5]

    Almousa and B

    A. Almousa and B. Lu. Ribbon complexes for the0-Hecke algebra, 2026. arXiv:2601.13324

    work page arXiv 2026

  6. [6]

    Benedetti and B

    C. Benedetti and B. E. Sagan. Antipodes and involutions.Journal of Combinatorial Theory, Series A, 148:275–315, 2017. doi: 10.1016/j.jcta.2016.12.005

    work page doi:10.1016/j.jcta.2016.12.005 2017

  7. [7]

    Duchamp, F

    G. Duchamp, F. Hivert, and J.-Y. Thibon. Noncommutative symmetric functions VI: free quasi- symmetric functions and related algebras.International Journal of Algebra and Computation, 12(5):671–717, 2002. doi: 10.1142/S0218196702001139

    work page doi:10.1142/s0218196702001139 2002

  8. [8]

    Ferroni, A

    L. Ferroni, A. H. Morales, and G. Panova. Skew shapes, Ehrhart positivity and beyond, 2025. To appear in Proceedings of the London Mathematical Society; arXiv:2503.16403

    arXiv 2025

Show all 40 references
  1. [9]

    B. Fresse. Iterated bar complexes ofE∞-algebras and homology theories.Algebraic & Geometric Topology, 11(2):747–838, 2011. doi: 10.2140/agt.2011.11.747

    2011 doi

  2. [10]

    I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon. Non- commutative symmetric functions.Advances in Mathematics, 112(2):218–348, 1995. doi: 10.1006/aima.1995.1032

    1995

  3. [11]

    I. M. Gessel and C. Reutenauer. Counting permutations with given cycle structure and descent set.Journal of Combinatorial Theory, Series A, 64(2):189–215, 1993. doi: 10.1016/ 0097-3165(93)90095-P

    1993

  4. [12]

    Hersh and S

    P. Hersh and S. Sundaram. Stability and ribbon bases for the rank-selected homology of geometric lattices, 2026. arXiv:2604.06479v2

    2026 arXiv

  5. [13]

    Hivert, J.-C

    F. Hivert, J.-C. Novelli, and J.-Y. Thibon. Commutative combinatorial Hopf algebras.Journal of Algebraic Combinatorics, 28(1):65–95, 2008. doi: 10.1007/s10801-007-0077-0

    2008 doi

  6. [14]

    P. C. Huang. Greedy records and Bernstein transfers for fence and circular-fence order polynomials, 2026. arXiv:2607.22767

    2026 arXiv

  7. [15]

    Jöllenbeck and V

    M. Jöllenbeck and V. Welker.Minimal Resolutions via Algebraic Discrete Morse Theory, volume 197 ofMemoirs of the American Mathematical Society, Number 923. American Mathematical Society, 2009. doi: 10.1090/memo/0923

    2009 doi

  8. [16]

    A. Joyal. Une théorie combinatoire des séries formelles.Advances in Mathematics, 42(1):1–82,

  9. [17]

    Y. Kahane. Combinatorial interpretation of the coefficients of the order polynomial of fence posets, 2026. arXiv:2607.11225

    2026 arXiv

  10. [18]

    Kreweras

    G. Kreweras. Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers.Cahiers du Bureau universitaire de recherche opérationnelle, Série Recherche, 6:9–107,

  11. [19]

    Loday and B

    J.-L. Loday and B. Vallette.Algebraic Operads, volume 346 ofGrundlehren der mathematischen Wissenschaften. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-30362-3

    2012 doi

  12. [20]

    I. G. Macdonald.Symmetric Functions and Hall Polynomials. Oxford Mathematical Mono- graphs. Clarendon Press, Oxford, second edition, 1995

    1995

  13. [21]

    Dualitybetweenquasi-symmetricalfunctionsandtheSolomon descent algebra.Journal of Algebra, 177(3):967–982, 1995

    C.MalvenutoandC.Reutenauer. Dualitybetweenquasi-symmetricalfunctionsandtheSolomon descent algebra.Journal of Algebra, 177(3):967–982, 1995. doi: 10.1006/jabr.1995.1336

    1995

  14. [22]

    Novelli, J.-Y

    J.-C. Novelli, J.-Y. Thibon, and F. Toumazet. A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions.Annals of Combinatorics, 24:557–576, 2020. doi: 10.1007/s00026-020-00504-5

    2020 doi

  15. [23]

    Ohsugi and A

    H. Ohsugi and A. Tsuchiya. Enriched chain polytopes.Israel Journal of Mathematics, 237(1): 485–500, 2020. doi: 10.1007/s11856-020-2012-1

    2020 doi

  16. [24]

    Ohsugi and A

    H. Ohsugi and A. Tsuchiya. Theh∗-polynomials of locally anti-blocking lattice polytopes and their γ-positivity.Discrete & Computational Geometry, 66(2):701–722, 2021. doi: 10.1007/ s00454-020-00236-6

    2021

  17. [25]

    I. Pak, B. Shapiro, I. Smirnov, and K.-i. Yoshida. Hilbert–Kunz multiplicity of quadrics via Ehrhart theory, 2025. arXiv:2508.17915

    2025

  18. [26]

    T. K. Petersen and Y. Zhuang. Zig-zag Eulerian polynomials.European Journal of Combina- torics, 124:104073, 2025. doi: 10.1016/j.ejc.2024.104073. 36 PYUYI CHUFENG HUANG

    2025

  19. [27]

    Romero and F

    A. Romero and F. Sergeraert. Discrete vector fields and fundamental algebraic topology, 2010. arXiv:1005.5685

    2010 arXiv

  20. [28]

    B. E. Sagan and S. Sundaram. Ordered set partition posets, 2026. To appear in Combinatorial Theory; arXiv:2506.23355v3

    2026 arXiv

  21. [29]

    W. R. Schmitt. Incidence Hopf algebras.Journal of Pure and Applied Algebra, 96(3):299–330,

  22. [30]

    Sköldberg

    E. Sköldberg. Morse theory from an algebraic viewpoint.Transactions of the American Mathematical Society, 358(1):115–129, 2006. doi: 10.1090/S0002-9947-05-04079-1

    2006 doi

  23. [31]

    L. Solomon. A decomposition of the group algebra of a finite Coxeter group.Journal of Algebra, 9(2):220–239, 1968. doi: 10.1016/0021-8693(68)90022-7

    1968 doi

  24. [32]

    R. P. Stanley. Some aspects of groups acting on finite posets.Journal of Combinatorial Theory, Series A, 32(2):132–161, 1982. doi: 10.1016/0097-3165(82)90017-6

    1982 doi

  25. [33]

    R. P. Stanley. Two poset polytopes.Discrete & Computational Geometry, 1(1):9–23, 1986. doi: 10.1007/BF02187680

    1986 doi

  26. [34]

    R. P. Stanley.Enumerative Combinatorics, Volume 1, volume 49 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, second edition, 2012. doi: 10.1017/ CBO9781139058520

    2012

  27. [35]

    R. P. Stanley.Enumerative Combinatorics, Volume 2, volume 208 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, second edition, 2024. doi: 10.1017/ 9781009262491

    2024

  28. [36]

    VandeBogert

    K. VandeBogert. Ribbon Schur functors.Algebra & Number Theory, 19(4):771–834, 2025. doi: 10.2140/ant.2025.19.771

    2025 doi

  29. [37]

    M. L. Wachs. Poset topology: tools and applications. InGeometric Combinatorics, volume 13 of IAS/Park City Mathematics Series, pages 497–615. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/pcms/013/09. School of Mathematics, Sichuan University, Chengdu 61006...

    2007 doi

  30. [1965]

    URLhttps://www.numdam.org/item/BURO_1965__6__9_0/

  31. [1981]

    doi: 10.1016/0001-8708(81)90052-9

    doi

  32. [1994]

    doi: 10.1016/0022-4049(94)90105-8

    doi

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