Matroid flat counts can have many peaks

Alexander Divoux; Chayim Lowen; Matt Larson; Shouda Wang

REVIEW 1 major objections 5 minor 300 references

Matroid flat counts can have many peaks

T0 review · 1 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Flat counts of matroids can have arbitrarily many peaks

desk verdict Solid disproof of Rota's conjecture with a clean construction, but the proof of the 'exactly m peaks' theorem leans on an unproven and in general false truncation claim that needs a small fix. read the letter →

arxiv 2608.07342 v1 pith:O6S6QRSA submitted 2026-08-07 math.CO

Alexander Divoux , Matt Larson , Chayim Lowen , Shouda Wang This is my paper · ORCID

classification math.CO MSC 05B35

keywords matroidWhitneynumbersofthesecondkindflatcountsunimodalitylog-concavityq-liftgeneralizedthetagraphcounterexample

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

Matroids are combinatorial structures that capture independence, and their flats—closed sets—can be counted by rank. This paper disproves the long-standing conjecture, posed at the 1970 International Congress of Mathematicians, that those rank-by-rank counts always rise to a single maximum and then fall. It proves the opposite: for every positive integer $m$, there is a matroid whose flat counts have exactly $m$ peaks. The construction finds generalized theta graphs where the counts fail log-concavity badly, takes direct sums to spread the failure across many ranks, and then applies the $q$-lift operation to turn those failures into strict peaks. The result settles the unimodality conjecture negatively and, as a direct consequence, also refutes the weaker log-concavity conjecture for these numbers.

What carries the argument

The $q$-lift is the central construction: given a simple matroid represented by vectors in $\mathbb{F}_q^r$, the lift adds one new vector and places scaled copies $(z,v)$ for all $z\in\mathbb{F}_q$, producing a rank-$(r+1)$ matroid over $\mathbb{F}_q$. Its flat counts obey the identity $\widetilde W_i=q^i W_i+W_{i-1}$, which is the mechanism that converts a severe failure of log-concavity in the original sequence into a strict peak after scaling by powers of $q$. The other half of the machinery is the asymptotic analysis of $G_t^{\oplus m}$, whose corank-$k$ counts have exponents $\lfloor 3(k+m)/2\rfloor$ in the middle range; these exponents are what locate the peaks.

What would settle it

Enumerate the flats of the rank-82 matroid from Example 3.2 and compare the exact counts at ranks 78, 79, and 80; if the middle count is not strictly larger than both neighbors, the claimed failure of unimodality for that example is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the Whitney numbers of the second kind—the counts $W_i$ of flats of rank $i$—need not form a unimodal sequence. The proof fixes $m$, takes the direct sum of $m$ copies of the graphic matroid of a generalized $\theta$ graph with four internally disjoint paths (three of length $t$ and one single edge), and forms a $p$-lift for a prime $p$ chosen so that $p=\Theta(t^{3/2})$. A quoted identity for $q$-lifts gives $\widetilde W_i=p^i W_i+W_{i-1}$ for the lifted flat counts, and asymptotic estimates for the direct sum show that the indices $m+2s+1$ for $0\le s\le m$ are strict peaks. Truncation then removes peaks one at a time, yielding exactly $m$ peaks for any prescribed $m$.

Load-bearing premise

The construction rests on the quoted formula for how many flats a $q$-lift has in each rank, and on the elementary, unproved fact that omitting the $W_1$ term from a sequence can destroy at most one peak.

Editorial extensions

If this is right

The unimodality conjecture for matroid flat counts is false, and in fact the number of peaks is unbounded.
The log-concavity conjecture for Whitney numbers of the second kind is also false, because any log-concave sequence with no internal zeros is unimodal.
For every positive integer $m$, there is an explicit matroid with exactly $m$ peaks; the method also gives a non-unimodal matroid on 4957 elements and a non-log-concave one on 78 elements.
Direct sums break log-concavity at many places simultaneously, so the failure is not confined to one rank in a specially chosen family.
The construction is representable over every field because the theta graphs are regular, so any future proof that flat counts are unimodal for representable matroids would have to leave these examples behind.

Reading between the lines

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

The same $q$-lift recipe should turn other families with known asymptotic exponents, such as the cothickened cocircuit examples mentioned in the paper, into further non-unimodal matroids that are not necessarily graphic.
The gap between the 78-element log-concavity failure and the 4957-element unimodality failure suggests the $q$-lift is the size bottleneck; using larger fields or theta graphs with more strands may shrink the examples substantially.
The method leaves the points-lines-planes inequality, which concerns only the first three entries of the flat-count sequence, untouched; a counterexample to that inequality, if one exists, would have to come from a different construction.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

This is the real thing: a disproof of Rota's 1970 conjecture that matroid flat counts are unimodal, and it goes further by showing the failure can be arbitrarily large. The two-step architecture is clean. Lemma 2.1 gives sharp asymptotic counts for the generalized theta graph matroid G_t; Lemma 2.4 computes the direct-sum convolution; and the q-lift argument converts the asymptotic log-concavity violations into strict local maxima. The Bonin–Qin formula is cited properly, and the use of Whittle's q-lift is careful about the representation-independence caveat. Example 3.2 even supplies an explicit 4957-element matroid with non-unimodal flat counts, which alone refutes the conjecture. The core asymptotic proof for 'arbitrarily many peaks' is solid and checkable.

The soft spot is the final truncation step in the proof of Theorem 1.2. After the q-lift construction, the paper has shown at least m+1 peaks. To get exactly m peaks, it invokes truncation and says 'it is clear that the number of peaks goes down by at most one' when the W_1 term is omitted. That claim is not true for arbitrary sequences: e.g., (1,5,5,4,6,1) has one peak (at the 6), while omitting the W_1 term gives (1,5,4,6,1), which has two peaks. So the blanket assertion fails. The paper doesn't supply the verification for its specific sequence. In this particular case the check does go through: apply m+1 truncations, the first peak at corank m+1 disappears, the remaining m peaks at m+3,...,3m+1 survive, and no new peak appears because the new boundary entry is a valley. But none of that appears in the text. As written, Theorem 1.2 is only proved for 'at least m+1 peaks,' not 'exactly m peaks,' unless the reader fills in the missing argument.

This is a genuine but small gap. It does not threaten the main counterexample to Rota's conjecture, which is established by Example 3.2 and by the arbitrarily-many-peaks construction. A referee should ask the authors to replace the hand-wave with a concrete truncation count. I'd send this to peer review and expect acceptance after that fix.

Who for: matroid theorists and combinatorial geometers; anyone tracking Mason's conjectures. I would cite it once the truncation step is repaired.

Regards.

Referee Report

1 major / 5 minor

Summary. The paper disproves Rota's conjecture that the Whitney numbers of the second kind of a matroid form a unimodal sequence. The main theorem, Theorem 1.2, asserts that for every positive integer m there is a matroid whose flat counts by rank have exactly m peaks. The proof combines asymptotic estimates for the flat counts of a family of generalized theta graphs G_t (Lemma 2.1), a convolution analysis for direct sums (Lemma 2.4), and Whittle's q-lift together with Bonin and Qin's formula for the flat counts of the lift (Lemma 3.1). The authors also exhibit concrete small matroids for which log-concavity and unimodality fail.

Significance. If the construction is correct, it refutes a fifty-year-old conjecture of Rota and, because log-concave sequences with no internal zeroes are unimodal, it also gives counterexamples to Mason's log-concavity conjecture for Whitney numbers of the second kind. The paper is largely self-contained and rests on published external results, namely Whittle's q-lift and Bonin-Qin's flat-count formula; the asymptotic arguments in Section 2 are carefully presented. The main weakness is the truncation step at the end of the proof of Theorem 1.2, which is asserted rather than proved; this step is load-bearing for the 'exactly m peaks' formulation, but the missing verification is elementary and local.

major comments (1)

[Section 3, proof of Theorem 1.2, final paragraph] The truncation step is load-bearing and is not proved. The proof establishes that the p-lift has at least m+1 peaks, and then asserts that truncation reduces the number of peaks by at most one, 'and that we can thereby achieve any desired number of peaks.' This assertion needs a proof: one must show that deleting a single entry from a sequence never increases the number of peaks (under the plateau convention of Definition 1), that the peak count eventually drops to 0 under repeated truncation, and that for the specific sequence constructed here the surviving old peaks remain strict local maxima and no new peaks appear at the boundary. Concretely, after k=m+1 truncations the corank sequence becomes (W_0, W_{m+2}, W_{m+3}, ..., W_r), and the old peaks at original coranks m+3, m+5, ..., 3m+1 survive; this verification is absent. Without it, the theorem as written only proves the existence of matroids with arbitrarily many peaks, not with a prescribed exact number.

minor comments (5)

[Section 1, paragraph after Theorem 1.2] The observation that a sequence is log-concave with no internal zeroes 'if and only if' every c-weighted sequence is unimodal is false as stated. For example, (1,2,5) is not log-concave, but for every c>0 the sequence (1, 2c, 5c^2) is increasing and hence unimodal. The proof of the main theorem does not use this claim, so the authors should remove it or replace it with a correct statement.
[Section 3, proof of Theorem 1.2, third display] In the estimate for p^{-ℓ} \tilde W_{m+2s+2}/W_{m+2s}, the term p^{-ℓ} O(t^3) is exponentially small because ℓ is linear in t, so writing the result as Θ(t^{-1/2}) is misleading. This should be stated as o(1) or O(t^{-1/2}) with an explicit note.
[Examples 2.2 and 3.2] The exact values of the Whitney numbers in these examples are stated without derivation or a reproducibility statement. Since the examples are used to support the 'smallest' claims, a computation outline or accompanying code would improve verifiability.
[Section 3, truncation paragraph] The sentence about omitting the W_1 term is only correct in corank indexing, which the paper introduced in Section 2 but does not repeat here. This should be clarified to avoid confusion with the rank-indexed W_i used in the introduction.
[References] The reference '[DL W26]' contains an unintended space in the author initials; it should read '[DLW26]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; the construction is self-contained and relies on external, non-self-cited results.

full rationale

The claimed derivation chain is not circular. The main theorem rests on (i) asymptotic flat-count estimates for generalized theta graphs and their direct sums, proved in Lemmas 2.1 and 2.4, and (ii) the external q-lift flat-count identity of Bonin and Qin, quoted as Lemma 3.1 from [BQ01, Lemma 4]. That identity is independent of this paper and is not derived from the target theorem, nor is it fitted to the data being predicted. The use of Whittle's q-lift and the regular representability of G_t^{oplus m} are also external, published facts. The only self-citations, [Lar26] and [DLW26], are explicitly described as recording AI-assisted initial examples and are not load-bearing for the proof. No parameter is fitted to a subset of the predicted quantities, and no 'prediction' reduces by construction to its input. One gap exists, but it is not a circularity: in the final paragraph of the proof of Theorem 1.2, the paper asserts, 'It is clear that the number of peaks goes down by at most one under such an omission and that we can thereby achieve any desired number of peaks.' This truncation step is asserted rather than proved; it is a potential correctness gap in the argument that the constructed matroid has exactly m peaks, rather than merely arbitrarily many peaks. But it is not an equation or citation that identifies the conclusion with an input, so it does not raise the circularity score. On the circularity axis, the paper is self-contained and scores 0.

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

The central theorem introduces no fitted constants: t is taken arbitrarily large, m is fixed, and p is any prime guaranteed by Bertrand's postulate. The specific strand lengths and q in Examples 2.2 and 3.2 are illustrative, not load-bearing. The paper invokes standard background results only and invents no new entities.

assumptions (6)

standard math Matroid duality rank formula: for any flat F of a matroid M, |E(M) \ F| - r_{M*}(E(M) \ F) = r(M) - r_M(F).
Used in Lemma 2.1 to count flats of G_t by counting cyclic sets of the dual G_t*.
standard math Bertrand's postulate: for every integer n>1 there is a prime p with n < p < 2n.
Used in Theorem 1.2 to choose p = Theta(t^{3/2}) so the p-lift exists.
standard math Whittle's q-lift construction yields a matroid for any simple F_q-representable matroid.
External result from [Whi89], used to build the lifted matroid in Section 3.
standard math Bonin-Qin flat-count formula for q-lifts: W_i(\tilde M) = q^i W_i(M) + W_{i-1}(M), and representation-independence of these numbers.
External result from [BQ01, Lemma 4], quoted as Lemma 3.1; it is the load-bearing identity in the peak construction.
standard math Flats of a direct sum of matroids are exactly unions of a flat from each summand, with rank equal to the sum of ranks.
Used in Lemma 2.4 to compute W_eta(G_t^{direct sum m}) via convolution.
standard math Graphic matroids are regular, hence representable over every field.
Used to ensure G_t^{direct sum m} is F_p-representable so the p-lift applies.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Matroid flat counts can have many peaks." pith.science (2026). https://pith.science/paper/O6S6QRSA

@misc{pith2026260807342,
  author       = {Pith},
  title        = {Pith review of: Matroid flat counts can have many peaks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6S6QRSA}},
  note         = {Machine review of arXiv:2608.07342}
}

abstract

We disprove Rota's conjecture that the counts of flats in a matroid according to rank form a unimodal sequence. Furthermore, we show that this sequence can have arbitrarily many peaks. The construction starts by finding a generalized theta graph for which log-concavity fails severely. By taking direct sums, we break log-concavity in many places. We then use Whittle's $q$-lift construction to produce a matroid whose flat counts have many peaks.

Figures

Figures reproduced from arXiv: 2608.07342 by the authors.

**Figure 1.** The generalized theta graph whose matroid is G28. coefficients of chromatic polynomials of graphs (and more generally characteristic polynomials of matroids) are known as Whitney numbers of the first kind, and they can be viewed as counts of flats weighted by the M¨obius function. However, it was shown in [AHK18] that Whitney numbers of the first kind are log-concave. The dual of the matroid Gt is a non-simple matr… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

300 extracted references · 105 canonical work pages

[1]

Aguiar, Marcelo and Ardila, Federico , TITLE =. Mem. Amer. Math. Soc. (to appear) , YEAR =
[2]

Discrete Comput

Ardila, Federico and Benedetti, Carolina and Doker, Jeffrey , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00454-009-9232-9 , URL =

doi:10.1007/s00454-009-9232-9 2010
[3]

Ardila, Federico and Castillo, Federico and Eur, Christopher and Postnikov, Alexander , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =

2020
[4]

Ardila, Federico and Denham, Graham and Huh, June , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2023 , NUMBER =. doi:10.1090/jams/1009 , URL =

doi:10.1090/jams/1009 2023
[5]

Valuations for matroid polytope subdivisions , JOURNAL =

Ardila, Federico and Fink, Alex and Rinc\'. Valuations for matroid polytope subdivisions , JOURNAL =. 2010 , NUMBER =. doi:10.4153/CJM-2010-064-9 , URL =

doi:10.4153/cjm-2010-064-9 2010
[6]

Adiprasito, Karim and Huh, June and Katz, Eric , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2018 , NUMBER =. doi:10.4007/annals.2018.188.2.1 , URL =

doi:10.4007/annals.2018.188.2.1 2018
[7]

Asgari, Mahdi , TITLE =. Canad. J. Math. , FJOURNAL =. 2002 , NUMBER =. doi:10.4153/CJM-2002-025-8 , URL =

work page doi:10.4153/cjm-2002-025-8 2002
[8]

Bastidas, Jose , TITLE =. Algebr. Comb. , FJOURNAL =. 2021 , NUMBER =

2021

Show all 300 references

[9]

Backman, Spencer and Eur, Christopher and Simpson, Connor , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2024 , NUMBER =. doi:10.4171/jems/1350 , URL =

2024 doi
[10]

Berget, Andrew and Eur, Christopher and Spink, Hunter and Tseng, Dennis , TITLE =. Invent. Math. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00222-023-01194-5 , URL =

2023 doi
[11]

Balla, George and Olarte, Jorge Alberto , title =
[12]

Duke Math

Berget, Andrew and Spink, Hunter and Tseng, Dennis , TITLE =. Duke Math. J. , FJOURNAL =. 2023 , NUMBER =. doi:10.1215/00127094-2023-0021 , URL =

2023 doi
[13]

, title=

Fink, Alex and Speyer, David E. , title=
[14]

Some combinatorial and algebraic properties of

Bj. Some combinatorial and algebraic properties of. Adv. Math. , FJOURNAL =. 1984 , NUMBER =. doi:10.1016/0001-8708(84)90021-5 , URL =

1984 doi
[15]

and Borovik, Alexandre V

Booth, Richard F. and Borovik, Alexandre V. and Gelfand, Israel and Stone, David A. , TITLE =. Ann. Comb. , FJOURNAL =. 2000 , NUMBER =

2000
[16]

Greedy algorithm and symmetric matroids , JOURNAL =

Bouchet, Andr\'. Greedy algorithm and symmetric matroids , JOURNAL =. 1987 , NUMBER =. doi:10.1007/BF02604639 , URL =

1987 doi
[17]

Multimatroids

Bouchet, Andr\'. Multimatroids. SIAM J. Discrete Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1137/S0895480193242591 , URL =

1997 doi
[18]

and Gelfand, Israel and White, Neil , TITLE =

Borovik, Alexandre V. and Gelfand, Israel and White, Neil , TITLE =. 2003 , PAGES =. doi:10.1007/978-1-4612-2066-4 , URL =

2003 doi
[19]

and Gelfand, Israel and White, Neil , TITLE =

Borovik, Alexandre V. and Gelfand, Israel and White, Neil , TITLE =. J. Algebraic Combin. , FJOURNAL =. 1998 , NUMBER =. doi:10.1023/A:1008610715466 , URL =

1998 doi
[20]

Lorentzian polynomials , JOURNAL =

Br\". Lorentzian polynomials , JOURNAL =. 2020 , NUMBER =. doi:10.4007/annals.2020.192.3.4 , URL =

2020 doi
[21]

and Proudfoot, Nicholas and Wang, Botong , TITLE =

Braden, Tom and Huh, June and Matherne, Jacob P. and Proudfoot, Nicholas and Wang, Botong , TITLE =. Adv. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.aim.2022.108646 , URL =

2022
[22]

Braden, Tom and Huh, June and Matherne, Jacob and Proudfoot, Nicholas and Wang, Botong , title=
[23]

European J

Brenti, Francesco , TITLE =. European J. Combin. , FJOURNAL =. 1994 , NUMBER =. doi:10.1006/eujc.1994.1046 , URL =

1994
[24]

Th\'. J. \'. 1837 , volume =
[25]

Brion, Michel , TITLE =. Ann. Sci. \'. 1988 , NUMBER =

1988
[26]

2010 , PAGES =

Chriss, Neil and Ginzburg, Victor , TITLE =. 2010 , PAGES =

2010
[27]

and Little, John B

Cox, David A. and Little, John B. and Schenck, Henry K. , TITLE =. 2011 , PAGES =. doi:10.1090/gsm/124 , URL =

2011 doi
[28]

, TITLE =

Crapo, Henry H. , TITLE =. Aequationes Math. , FJOURNAL =. 1969 , PAGES =

1969
[29]

European J

Crapo, Henry and Schmitt, William , TITLE =. European J. Combin. , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.ejc.2004.05.005 , URL =

2005 doi
[30]

Cameron, Amanda and Fink, Alex , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.jcta.2021.105584 , URL =

2022
[31]

and Rueckriemen, Ralf , TITLE =

Chun, Carolyn and Moffatt, Iain and Noble, Steven D. and Rueckriemen, Ralf , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.jcta.2019.02.023 , URL =

2019 doi
[32]

and Rueckriemen, Ralf , TITLE =

Chun, Carolyn and Moffatt, Iain and Noble, Steven D. and Rueckriemen, Ralf , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/plms.12190 , URL =

2019 doi
[33]

, TITLE =

Danielsen, Lars Eirik and Parker, Matthew G. , TITLE =. Discrete Appl. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.dam.2009.11.011 , URL =

2010 doi
[34]

1995 , NUMBER =

Wonderful models of subspace arrangements , JOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01589496 , URL =

1995 doi
[35]

Derksen, Harm and Fink, Alex , TITLE =. Adv. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.aim.2010.04.016 , URL =

2010 doi
[36]

Dolgachev, Igor and Lunts, Valery , TITLE =. J. Algebra , FJOURNAL =. 1994 , NUMBER =

1994
[37]

Combinatorial

Edmonds, Jack , TITLE =. Combinatorial. 1970 , MRCLASS =

1970
[38]

Forum Math

Eur, Christopher and Huh, June and Larson, Matt , TITLE =. Forum Math. Pi , FJOURNAL =. 2023 , PAGES =. doi:10.1017/fmp.2023.24 , URL =

2023 doi
[39]

Eur, Christopher and Sanchez, Mario and Supina, Mariel , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2021 , NUMBER =

2021
[40]

1993 , pages=

Fulton, William , title=. 1993 , pages=

1993
[41]

Topology , volume=

Intersection theory on toric varieties , author=. Topology , volume=. 1997 , publisher=

1997
[42]

1874 , volume =

Om rumvinklerne i et polyeder , journal =. 1874 , volume =
[43]

Uspekhi Mat

Gelfand, Israel and Serganova, Vera , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1987 , NUMBER =

1987
[44]

Hampe, Simon , TITLE =. J. Combin. Theory Ser. B , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.jctb.2016.08.004 , URL =

2017 doi
[45]

1977 , PAGES =

Hartshorne, Robin , TITLE =. 1977 , PAGES =

1977
[46]

Current Developments in Mathematics 2016 , pages=

Huh, June , title=. Current Developments in Mathematics 2016 , pages=

2016
[47]

Huh, June and Katz, Eric , TITLE =. Math. Ann. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00208-011-0777-6 , URL =

2012 doi
[48]

The interlace polynomial of a graph , JOURNAL =

Arratia, Richard and Bollob\'. The interlace polynomial of a graph , JOURNAL =. 2004 , NUMBER =. doi:10.1016/j.jctb.2004.03.003 , URL =

2004 doi
[49]

European J

Brijder, Robert and Hoogeboom, Hendrik Jan , TITLE =. European J. Combin. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.ejc.2014.03.005 , URL =

2014 doi
[50]

Graph polynomials , SERIES =

Morse, Ada , TITLE =. Graph polynomials , SERIES =. 2017 , MRCLASS =

2017
[51]

Linear Algebra Appl

Aigner, Martin and van der Holst, Hein , TITLE =. Linear Algebra Appl. , FJOURNAL =. 2004 , PAGES =. doi:10.1016/j.laa.2003.06.010 , URL =

2004 doi
[52]

Trudy Mat

Klyachko, Alexander , TITLE =. Trudy Mat. Inst. Steklov. , FJOURNAL =. 1995 , NUMBER =

1995
[53]

Lenz, Matthias , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1016/j.aam.2013.07.001 , URL =

2013 doi
[54]

L\'. Chern-. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2020 , NUMBER =

2020
[55]

McMullen, Peter , TITLE =. Adv. Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1016/0001-8708(89)90029-7 , URL =

1989 doi
[56]

McMullen, Peter , TITLE =. Adv. Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.aim.2008.09.004 , URL =

2009 doi
[57]

2005 , series=

Miller, Ezra and Sturmfels, Bernd , title=. 2005 , series=

2005
[58]

Morelli, Robert , TITLE =. Adv. Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.1006/aima.1993.1032 , URL =

1993
[59]

2015 , pages=

Maclagan, Diane and Sturmfels, Bernd , title=. 2015 , pages=

2015
[60]

Padrol, Arnau and Pilaud, Vincent and Ritter, Julian , title =
[61]

Postnikov, Alexander , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2009 , NUMBER =. doi:10.1093/imrn/rnn153 , URL =

2009 doi
[62]

, TITLE =

Procesi, C. , TITLE =. Mots , SERIES =. 1990 , MRCLASS =

1990
[63]

, TITLE =

Proudfoot, Nicholas and Speyer, David E. , TITLE =. Beitr\". 2006 , NUMBER =

2006
[64]

Isotropical linear spaces and valuated

Rinc\'. Isotropical linear spaces and valuated. J. Combin. Theory Ser. A , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.jcta.2011.08.001 , URL =

2012 doi
[65]

2003 , PAGES =

Schrijver, Alexander , TITLE =. 2003 , PAGES =

2003
[66]

Scholten, Georgy and Vinzant, Cynthia , TITLE =. Algebr. Comb. , FJOURNAL =. 2019 , NUMBER =. doi:10.5802/alco.65 , URL =

2019 doi
[67]

Topics in algebraic geometry and geometric modeling , SERIES =

Sottile, Frank , TITLE =. Topics in algebraic geometry and geometric modeling , SERIES =. 2003 , MRCLASS =. doi:10.1090/conm/334/05984 , URL =

2003 doi
[68]

, TITLE =

Stanley, Richard P. , TITLE =. 2012 , PAGES =

2012
[69]

, TITLE =

Stembridge, John R. , TITLE =. Adv. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1006/aima.1994.1058 , URL =

1994
[70]

Terao, Hiroaki , TITLE =. J. Algebra , FJOURNAL =. 2002 , NUMBER =

2002
[71]

and Williams, L

Tsukerman, E. and Williams, L. , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =

2015
[72]

Tutte, W. T. , TITLE =. J. Combinatorial Theory , FJOURNAL =. 1967 , PAGES =

1967
[73]

Compositio Math

Varchenko, Alexander , TITLE =. Compositio Math. , FJOURNAL =. 1995 , NUMBER =

1995
[74]

Linear Algebra Appl

Duchamp, Alain , TITLE =. Linear Algebra Appl. , FJOURNAL =. 1992 , PAGES =. doi:10.1016/0024-3795(92)90441-C , URL =

1992 doi
[75]

Br. Isr. Jour. Math. (to appear) , year=
[76]

Edidin, Dan and Graham, William , TITLE =. Invent. Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/s002220050214 , URL =

1998 doi
[77]

Payne, Sam , TITLE =. Math. Res. Lett. , FJOURNAL =. 2006 , NUMBER =. doi:10.4310/MRL.2006.v13.n1.a3 , URL =

2006 doi
[78]

Vezzosi, Gabriele and Vistoli, Angelo , TITLE =. Invent. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00222-002-0275-2 , URL =

2003 doi
[79]

2011 , PAGES =

Oxley, James , TITLE =. 2011 , PAGES =. doi:10.1093/acprof:oso/9780198566946.001.0001 , URL =

2011
[80]

Manuscripta Math

Clader, Emily and Damiolini, Chiara and Huang, Daoji and Li, Shiyue and Ramadas, Rohini , TITLE =. Manuscripta Math. , YEAR =
[81]

Discrete Optim

Fujishige, Satoru , TITLE =. Discrete Optim. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.disopt.2014.02.002 , URL =

2014 doi
[82]

Discrete Appl

Fujishige, Satoru , TITLE =. Discrete Appl. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.dam.2017.04.047 , URL =

2017 doi
[83]

Fujishige, Satoru and Patkar, Sachin , Title =
[84]

Ferroni, Luis and Schr\"oter, Benjamin , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/jlms.12984 , URL =

2024 doi
[85]

Eur, Christopher and Fink, Alex and Larson, Matt and Spink, Hunter , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/plms.12592 , URL =

2024 doi
[86]

and Sokal, Alan D

Choe, Young-Bin and Oxley, James G. and Sokal, Alan D. and Wagner, David G. , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/S0196-8858(03)00078-2 , URL =

2004 doi
[87]

Internat

Ishida, Masa-Nori , TITLE =. Internat. J. Math. , FJOURNAL =. 1990 , NUMBER =

1990
[88]

Hibi, Takayuki , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1989 , NUMBER =. doi:10.1016/0097-3165(89)90025-3 , URL =

1989 doi
[89]

European J

Morse, Ada , TITLE =. European J. Combin. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.ejc.2019.01.005 , URL =

2019 doi
[90]

The homology and shellability of matroids and geometric lattices , BOOKTITLE =

Bj\". The homology and shellability of matroids and geometric lattices , BOOKTITLE =. 1992 , MRCLASS =. doi:10.1017/CBO9780511662041.008 , URL =

1992 doi
[91]

Nima Anari and Kuikui Liu and Shayan Oveis Gharan and Cynthia Vinzant , title =
[92]

Dunstan, F. D. J. and Welsh, Dominic , TITLE =. Math. Programming , FJOURNAL =. 1973 , PAGES =. doi:10.1007/BF01580137 , URL =

1973 doi
[93]

Fujishige, Satoru and Iwata, Satoru , TITLE =. SIAM J. Discrete Math. , FJOURNAL =. 2005 , NUMBER =. doi:10.1137/S0895480103426339 , URL =

2005 doi
[94]

Delta-matroids, jump systems, and bisubmodular polyhedra , JOURNAL =

Bouchet, Andr\'. Delta-matroids, jump systems, and bisubmodular polyhedra , JOURNAL =. 1995 , NUMBER =. doi:10.1137/S0895480191222926 , URL =

1995 doi
[95]

and Kabadi, Santosh N

Chandrasekaran, R. and Kabadi, Santosh N. , TITLE =. Discrete Math. , FJOURNAL =. 1988 , NUMBER =. doi:10.1016/0012-365X(88)90101-X , URL =

1988 doi
[96]

Representability of

Bouchet, Andr\'. Representability of. Combinatorics (. 1988 , MRCLASS =

1988
[97]

Wenzel, Walter , TITLE =. Appl. Math. Lett. , FJOURNAL =. 1993 , NUMBER =. doi:10.1016/0893-9659(93)90103-T , URL =

1993 doi
[98]

Mason, J. H. , TITLE =. Combinatorics (. 1972 , MRCLASS =

1972
[99]

, TITLE =

Fink, Alex and Speyer, David E. , TITLE =. Duke Math. J. , FJOURNAL =. 2012 , NUMBER =. doi:10.1215/00127094-1813296 , URL =

2012 doi
[100]

Dinu, Rodica and Eur, Christopher and Seynnaeve, Tim , TITLE =. ar
[101]

Flag matroids: algebra and geometry , BOOKTITLE =

Cameron, Amanda and Dinu, Rodica and Micha. Flag matroids: algebra and geometry , BOOKTITLE =. 2022 , MRCLASS =. doi:10.1007/978-3-030-98327-7\_4 , URL =

2022 doi
[102]

, TITLE =

Speyer, David E. , TITLE =. Adv. Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.aim.2009.01.010 , URL =

2009 doi
[103]

Welsh, D. J. A. , Date-Added =. Matroid theory , Url =. 1976 , Bdsk-Url-1 =

1976
[104]

Shadows of characteristic cycles,

Aluffi, Paolo and Mihalcea, Leonardo C and Sch. Shadows of characteristic cycles,. To appear in
[105]

, TITLE =

Funk, Daryl and Mayhew, Dillon and Noble, Steven D. , TITLE =. European J. Combin. , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.ejc.2017.10.005 , URL =

2018 doi
[106]

Crowley, Colin and Huh, June and Larson, Matt and Simpson, Connor and Wang, Botong , title =
[107]

Larson, Matt and Payne, Sam and Stapledon, Alan , title =
[108]

Larson, Matt and Li, Shiyue and Payne, Sam and Proudfoot, Nicholas , TITLE =. Adv. Math. , FJOURNAL =. 2024 , PAGES =. doi:10.1016/j.aim.2024.109554 , URL =

2024
[109]

Eur, Christopher and Larson, Matt , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2024 , NUMBER =. doi:10.1093/imrn/rnad213 , URL =

2024 doi
[110]

Ferroni, Luis and Larson, Matt , TITLE =. Commun. Am. Math. Soc. , FJOURNAL =. 2024 , PAGES =. doi:10.1090/cams/28 , URL =

2024 doi
[111]

Larson, Matt , title =
[112]

Alon, Noga and Kravitz, Noah and Larson, Matt , TITLE =. J. Number Theory , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.jnt.2020.10.009 , URL =

2021 doi
[113]

Gao, Alice L. L. and Xie, Matthew H. Y. , TITLE =. J. Combin. Theory Ser. B , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.jctb.2021.07.004 , URL =

2021 doi
[114]

Gao, Alice L. L. and Lu, Linyuan and Xie, Matthew H. Y. and Yang, Arthur L. B. and Zhang, Philip B. , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.aam.2020.102117 , URL =

2021
[115]

Ferroni, Luis and Vecchi, Lorenzo , TITLE =. Algebr. Comb. , FJOURNAL =. 2022 , NUMBER =. doi:10.5802/alco.244 , URL =

2022 doi
[116]

Electron

Braden, Tom and Vysogorets, Artem , TITLE =. Electron. J. Combin. , FJOURNAL =. 2020 , NUMBER =

2020
[117]

Sloane, Neil J. A. , TITLE =. Notices Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =

2018
[118]

Algebra Number Theory , FJOURNAL =

Tosteson, Philip , TITLE =. Algebra Number Theory , FJOURNAL =. 2022 , NUMBER =. doi:10.2140/ant.2022.16.2433 , URL =

2022 doi
[119]

Gedeon, Katie and Proudfoot, Nicholas and Young, Benjamin , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.jcta.2017.03.007 , URL =

2017 doi
[120]

Gedeon, Katie and Proudfoot, Nicholas and Young, Benjamin , TITLE =. S\'. 2017 , PAGES =

2017
[121]

Elias, Ben and Proudfoot, Nicholas and Wakefield, Max , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2016.05.005 , URL =

2016 doi
[122]

, TITLE =

Oxley, James G. , TITLE =. Stud. Appl. Math. , FJOURNAL =. 1982 , NUMBER =. doi:10.1002/sapm1982663267 , URL =

1982 doi
[123]

, TITLE =

Brylawski, Thomas H. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1971 , PAGES =. doi:10.2307/1995423 , URL =

1971 doi
[124]

2008 , PAGES =

Drake, Brian , TITLE =. 2008 , PAGES =

2008
[125]

doi:10.48550/arXiv.2010.06088v2 , archivePrefix =

Singular H odge theory for combinatorial geometries. doi:10.48550/arXiv.2010.06088v2 , archivePrefix =. 2010.06088v3 , primaryClass =

2010 doi arXiv
[126]

arXiv e-prints , keywords =

Hilbert-Poincar \'e series of matroid Chow rings and intersection cohomology. arXiv e-prints , keywords =. doi:10.48550/arXiv.2212.03190v2 , archivePrefix =. 2212.03190v2 , primaryClass =

doi arXiv
[127]

New York J

Proudfoot, Nicholas and Young, Ben , TITLE =. New York J. Math. , FJOURNAL =. 2017 , PAGES =

2017
[128]

Xie, Matthew H. Y. and Zhang, Philip B. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/proc/14608 , URL =

2019 doi
[129]

Electron

Proudfoot, Nicholas and Xu, Yuan and Young, Ben , TITLE =. Electron. J. Combin. , FJOURNAL =. 2018 , NUMBER =

2018
[130]

Proudfoot, Nicholas , TITLE =. Algebr. Comb. , FJOURNAL =. 2021 , NUMBER =. doi:10.5802/alco.174 , URL =

2021 doi
[131]

Proudfoot, Nicholas , TITLE =. Algebr. Comb. , FJOURNAL =. 2019 , NUMBER =. doi:10.5802/alco.59 , URL =

2019 doi
[132]

, TITLE =

Gedeon, Katie R. , TITLE =. Electron. J. Combin. , FJOURNAL =. 2017 , NUMBER =

2017
[133]

Proudfoot, Nicholas and Wakefield, Max and Young, Ben , TITLE =. J. Algebraic Combin. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s10801-015-0628-8 , URL =

2016 doi
[134]

and Radcliffe, Jamie , TITLE =

Lee, Kyungyong and Nasr, George D. and Radcliffe, Jamie , TITLE =. Electron. J. Combin. , FJOURNAL =. 2021 , NUMBER =. doi:10.37236/10602 , URL =

2021 doi
[135]

Lu, Linyuan and Xie, Matthew H. Y. and Yang, Arthur L. B. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.jcta.2022.105665 , URL =

2022
[136]

, TITLE =

Kapranov, Michael M. , TITLE =. I. 1993 , MRCLASS =

1993
[137]

Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , issn =. doi:10.1093/imrn/rnac270 , url =

2022 doi
[138]

, TITLE =

Crapo, Henry H. , TITLE =. J. Combinatorial Theory , FJOURNAL =. 1967 , PAGES =

1967
[139]

Seymour, P. D. , TITLE =. Handbook of combinatorics,. 1995 , MRCLASS =

1995
[140]

2009 , PAGES =

Flajolet, Philippe and Sedgewick, Robert , TITLE =. 2009 , PAGES =. doi:10.1017/CBO9780511801655 , URL =

2009 doi
[141]

, TITLE =

Stanley, Richard P. , TITLE =. 1999 , PAGES =. doi:10.1017/CBO9780511609589 , URL =

1999 doi
[142]

and Wakefield, Max D

Karn, Trevor K. and Wakefield, Max D. , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aam.2018.09.003 , URL =

2019 doi
[143]

, TITLE =

Moon, John W. , TITLE =. Random graphs '85 (. 1987 , MRCLASS =

1987
[144]

Enumerations, forbidden subgraph characterizations, and the split-decomposition , JOURNAL =

Bahrani, Maryam and Lumbroso, J\'. Enumerations, forbidden subgraph characterizations, and the split-decomposition , JOURNAL =. 2018 , NUMBER =. doi:10.37236/6431 , URL =

2018 doi
[145]

Welsh, D. J. A. , TITLE =. Combinatorial. 1971 , MRCLASS =

1971
[146]

Tohoku Math

Barthel, Gottfried and Brasselet, Jean-Paul and Fieseler, Karl-Heinz and Kaup, Ludger , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 2002 , NUMBER =

2002
[147]

, TITLE =

Bressler, Paul and Lunts, Valery A. , TITLE =. Compositio Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1023/A:1022232232018 , URL =

2003 doi
[148]

Karu, Kalle , TITLE =. Invent. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00222-004-0358-3 , URL =

2004 doi
[149]

, TITLE =

Stanley, Richard P. , TITLE =. Commutative algebra and combinatorics (. 1987 , MRCLASS =. doi:10.2969/aspm/01110187 , URL =

1987
[150]

, TITLE =

Stanley, Richard P. , TITLE =. Adv. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.1016/0001-8708(80)90050-X , URL =

1980 doi
[151]

Huh, June , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0894-0347-2012-00731-0 , URL =

2012 doi
[152]

Huh, June , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.aim.2014.11.002 , URL =

2015 doi
[153]

Feichtner, Eva Maria and Yuzvinsky, Sergey , TITLE =. Invent. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00222-003-0327-2 , URL =

2004 doi
[154]

Berget, Andrew and Fink, Alex , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/imrn/rnab135 , URL =

2022 doi
[155]

Brion, Michel , TITLE =. J. Algebra , FJOURNAL =. 2002 , NUMBER =. doi:10.1016/S0021-8693(02)00505-7 , URL =

2002 doi
[156]

, TITLE =

Wagner, David G. , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1006/aama.1998.0610 , URL =

1998
[157]

Shapiro, Boris and Smirnov, Ilya and Vaintrob, Arkady , title =
[158]

Pagaria, Roberto and Pezzoli, Gian Marco , title =
[159]

Eur, Christopher and Larson, Matt and Spink, Hunter , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/tran/9328 , URL =

2025 doi
[160]

Commutative algebra (

Brion, Michel , TITLE =. Commutative algebra (. 2003 , ISBN =. doi:10.1090/conm/331/05900 , URL =

2003 doi
[161]

Bloch, Spencer and Gieseker, David , TITLE =. Invent. Math. , FJOURNAL =. 1971 , PAGES =. doi:10.1007/BF01404655 , URL =

1971 doi
[162]

Internat

Lee, Yuan-Pin , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 1997 , NUMBER =. doi:10.1155/S1073792897000263 , URL =

1997 doi
[163]

Silversmith, Rob , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/proc/16016 , URL =

2022 doi
[164]

Lovett, Shachar , TITLE =. 59th. 2018 , MRCLASS =. doi:10.1109/FOCS.2018.00027 , URL =

2018
[165]

IEEE Trans

Yildiz, Hikmet and Hassibi, Babak , TITLE =. IEEE Trans. Inform. Theory , FJOURNAL =. 2019 , NUMBER =. doi:10.1109/TIT.2019.2932663 , URL =

2019
[166]

, TITLE =

Stanley, Richard P. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1992 , NUMBER =. doi:10.2307/2152711 , URL =

1992 doi
[167]

Ferroni, Luis , TITLE =. Comb. Theory , FJOURNAL =. 2023 , NUMBER =

2023
[168]

Brakensiek, Joshua and Eur, Christopher and Larson, Matt and Li, Shiyue , title =
[169]

Caminata, Alessio and Cid-Ruiz, Yairon and Conca, Aldo , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.109361 , URL =

2023
[170]

K-polynomials of multiplicity-free varieties , note =

Castillo, Federico and Cid-Ruiz, Yairon and Mohammadi, Fatemeh and Monta\. K-polynomials of multiplicity-free varieties , note =
[171]

2005 , PAGES =

Brion, Michel and Kumar, Shrawan , TITLE =. 2005 , PAGES =

2005
[172]

2004 , PAGES =

Lazarsfeld, Robert , TITLE =. 2004 , PAGES =. doi:10.1007/978-3-642-18808-4 , URL =

2004 doi
[173]

1972 , booktitle=

Thorkell Helgason , title=. 1972 , booktitle=

1972
[174]

Colin McDiarmid , title=. Math. Proc. Camb. Phil. Soc. , year=
[175]

Flats in matroids and geometric graphs , BOOKTITLE =

Lov\'. Flats in matroids and geometric graphs , BOOKTITLE =. 1977 , MRCLASS =

1977
[176]

Nguyen , title=

Hien Q. Nguyen , title=. Theory of matroids , chapter=10, editor=
[177]

Theory of matroids , SERIES =

Crapo, Henry , TITLE =. Theory of matroids , SERIES =. 1986 , ISBN =

1986
[178]

and Chun, Carolyn and Fife, Tara , title =

Bonin, Joseph E. and Chun, Carolyn and Fife, Tara , title =
[179]

Frobenius splitting and

Knutson, Allen , note=. Frobenius splitting and
[180]

Brion, Michel , TITLE =. Comment. Math. Helv. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/PL00000379 , URL =

2001 doi
[181]

1998 , PAGES =

Fulton, William , TITLE =. 1998 , PAGES =. doi:10.1007/978-1-4612-1700-8 , URL =

1998 doi
[182]

1995 , PAGES =

Eisenbud, David , TITLE =. 1995 , PAGES =. doi:10.1007/978-1-4612-5350-1 , URL =

1995 doi
[183]

, TITLE =

Stanley, Richard P. , TITLE =. 1996 , PAGES =

1996
[184]

Eur, Christopher and Larson, Matt , TITLE =. Algebr. Geom. , FJOURNAL =. 2026 , NUMBER =

2026
[185]

Pollaczek-Geiringer, Hilda , title =. Z.Angew. Math. Mech. , year =
[186]

Laman, Gerard , TITLE =. J. Engrg. Math. , FJOURNAL =. 1970 , PAGES =. doi:10.1007/BF01534980 , URL =

1970 doi
[187]

Electron

Gallet, Matteo and Grasegger, Georg and Schicho, Josef , TITLE =. Electron. J. Combin. , FJOURNAL =. 2020 , NUMBER =. doi:10.37236/8548 , URL =

2020 doi
[188]

, TITLE =

McMullen, P. , TITLE =. Israel J. Math. , FJOURNAL =. 1971 , PAGES =. doi:10.1007/BF02771471 , URL =

1971 doi
[189]

Denef, Jan and Loeser, Fran. Motivic. J. Algebraic Geom. , FJOURNAL =. 1998 , NUMBER =

1998
[190]

Bultot, Emmanuel and Nicaise, Johannes , TITLE =. Math. Z. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00209-019-02342-5 , URL =

2020 doi
[191]

Bories, Bart and Veys, Willem , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2016 , NUMBER =. doi:10.1090/memo/1145 , URL =

2016 doi
[192]

Lemahieu, Ann and Van Proeyen, Lise , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9947-2011-05301-8 , URL =

2011 doi
[193]

Larson, Matt and Payne, Sam and Stapledon, Alan , TITLE =. Algebr. Comb. , FJOURNAL =. 2023 , NUMBER =. doi:10.5802/alco.293 , URL =

2023 doi
[194]

Ardila, Federico and Postnikov, Alexander , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0002-9947-10-05018-X , URL =

2010 doi
[195]

, TITLE =

Fink, Alex and Shaw, Kris and Speyer, David E. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2025 , NUMBER =. doi:10.1093/imrn/rnaf329 , URL =

2025 doi
[196]

2024 , note =

Berget, Andrew and Fink, Alex , title =. 2024 , note =

2024
[197]

Ardila, Federico and Boocher, Adam , TITLE =. J. Algebraic Combin. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s10801-015-0634-x , URL =

2016 doi
[198]

Bernstein, Daniel Irving , TITLE =. SIAM J. Appl. Algebra Geom. , FJOURNAL =. 2022 , NUMBER =. doi:10.1137/20M1346961 , URL =

2022 doi
[199]

Ardila, Federico and Postnikov, Alexander , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S0002-9947-2015-06071-1 , URL =

2015 doi
[200]

Berget, Andrew and Fink, Alex , TITLE =. Beitr. Algebra Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s13366-018-0402-x , URL =

2018 doi
[201]

Reineke, Markus and Rhoades, Brendon and Tewari, Vasu , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnad033 , URL =

2023 doi
[202]

2024 , note =

Rhoades, Brendon and Tewari, Vasu and Wilson, Andy , title =. 2024 , note =

2024
[203]

Holtz, Olga and Ron, Amos , TITLE =. Adv. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.aim.2011.02.012 , URL =

2011 doi
[204]

Sturmfels, Bernd and Xu, Zhiqiang , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2010 , NUMBER =. doi:10.4171/JEMS/204 , URL =

2010 doi
[205]

Lenz, Matthias , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2016.03.005 , URL =

2016 doi
[206]

Holtz, Olga and Ron, Amos and Xu, Zhiqiang , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0002-9947-2011-05329-8 , URL =

2012 doi
[207]

European J

Berget, Andrew , TITLE =. European J. Combin. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.ejc.2010.01.006 , URL =

2010 doi
[208]

Crowley, Colin and Proudfoot, Nicholas , title =
[209]

The geometry of zonotopal algebras

Crowley, Colin and Dorpalen-Barry, Galen and Henriques, Andr\'. The geometry of zonotopal algebras
[210]

Angarone, Robert and Commins, Patricia and Karn, Trevor and Murai, Satoshi and Rhoades, Brendon , TITLE =. Adv. Math. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.aim.2025.110185 , URL =

2025
[211]

Forum Math

Rhoades, Brendon and Wilson, Andy , TITLE =. Forum Math. Pi , FJOURNAL =. 2024 , PAGES =. doi:10.1017/fmp.2024.14 , URL =

2024 doi
[212]

Murai, Satoshi and Rhoades, Brendon and Wilson, Andy , title =
[213]

Kim, Jongwon and Rhoades, Brendon , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/imrn/rnaa203 , URL =

2022 doi
[214]

Forum Math

Rhoades, Brendon and Wilson, Andrew Timothy , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.90 , URL =

2022 doi
[215]

2003 , PAGES =

Weyman, Jerzy , TITLE =. 2003 , PAGES =. doi:10.1017/CBO9780511546556 , URL =

2003 doi
[216]

, TITLE =

Kempf, George R. , TITLE =. Invent. Math. , FJOURNAL =. 1976 , NUMBER =. doi:10.1007/BF01390321 , URL =

1976 doi
[217]

Hodge ideals , JOURNAL =

Musta. Hodge ideals , JOURNAL =. 2019 , NUMBER =. doi:10.1090/memo/1268 , URL =

2019 doi
[218]

Algebraic threefolds (

Esnault, H\'el\`ene and Viehweg, Eckart , TITLE =. Algebraic threefolds (. 1982 , ISBN =

1982
[219]

Differential topology, infinite-dimensional

Postnikov, Alexander and Shapiro, Boris and Shapiro, Mikhail , TITLE =. Differential topology, infinite-dimensional. 1999 , ISBN =. doi:10.1090/trans2/194/10 , URL =

1999 doi
[220]

, TITLE =

Dahmen, Wolfgang and Micchelli, Charles A. , TITLE =. Studia Math. , FJOURNAL =. 1985 , NUMBER =. doi:10.4064/sm-82-3-243-263 , URL =

1985 doi
[221]

Brion, Michel and Vergne, Mich\`ele , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1999 , NUMBER =. doi:10.1016/S0012-9593(01)80005-7 , URL =

1999 doi
[222]

2025 , PAGES =

Vakil, Ravi , TITLE =. 2025 , PAGES =

2025
[223]

, TITLE =

Olsson, Martin C. , TITLE =. Math. Ann. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00208-005-0707-6 , URL =

2005 doi
[224]

, TITLE =

Stanley, Richard P. , TITLE =. European J. Combin. , FJOURNAL =. 1993 , NUMBER =. doi:10.1006/eujc.1993.1028 , URL =

1993
[225]

, TITLE =

Stanley, Richard P. , TITLE =. Ann. Discrete Math. , FJOURNAL =. 1980 , PAGES =

1980
[226]

, TITLE =

Stanley, Richard P. , TITLE =. Studies in Appl. Math. , FJOURNAL =. 1975 , NUMBER =. doi:10.1002/sapm1975542135 , URL =

1975 doi
[227]

Speyer, David and Sturmfels, Bernd , TITLE =. Adv. Geom. , FJOURNAL =. 2004 , NUMBER =. doi:10.1515/advg.2004.023 , URL =

2004 doi
[228]

, TITLE =

Speyer, David E. , TITLE =. SIAM J. Discrete Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1137/080716219 , URL =

2008 doi
[229]

Totaro, Burt , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/jag/724 , URL =

2019 doi
[230]

1997 , PAGES =

Fulton, William , TITLE =. 1997 , PAGES =

1997
[231]

Smirnov, A. L. , TITLE =. Algebra i Analiz , FJOURNAL =. 1996 , NUMBER =

1996
[232]

1993 , PAGES =

Bruns, Winfried and Herzog, J\"urgen , TITLE =. 1993 , PAGES =

1993
[233]

2024 , PAGES =

Bocci, Cristiano and Carlini, Enrico , TITLE =. 2024 , PAGES =. doi:10.1007/978-3-031-54263-3 , URL =

2024 doi
[234]

Algebraic and geometric combinatorics on lattice polytopes , PAGES =

Olarte, Jorge Alberto and Panizzut, Marta and Schr\"oter, Benjamin , TITLE =. Algebraic and geometric combinatorics on lattice polytopes , PAGES =. 2019 , ISBN =

2019
[235]

Chatelain, Vanessa and Ram\'irez Alfons\'in, Jorge Luis , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.aam.2010.04.005 , URL =

2011 doi
[236]

European J

Beck, Matthias and Sottile, Frank , TITLE =. European J. Combin. , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.ejc.2005.06.003 , URL =

2007 doi
[237]

Gelfand, I. M. and Goresky, R. M. and MacPherson, R. D. and Serganova, V. V. , TITLE =. Adv. in Math. , FJOURNAL =. 1987 , NUMBER =. doi:10.1016/0001-8708(87)90059-4 , URL =

1987 doi
[238]

Chatzistamatiou, Andre and R\"ulling, Kay , TITLE =. Compos. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1112/S0010437X15007435 , URL =

2015 doi
[239]

2024 , PAGES =

Larson, Matt , TITLE =. 2024 , PAGES =

2024
[240]

1971 , PAGES =

Grothendieck, Alexander , TITLE =. 1971 , PAGES =

1971
[241]

, TITLE =

Smith, Karen E. , TITLE =. Algebraic geometry---. 1997 , ISBN =. doi:10.1090/pspum/062.1/1492526 , URL =

1997 doi
[242]

Li, Binglin , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2018 , NUMBER =. doi:10.1093/imrn/rnx003 , URL =

2018 doi
[243]

Castillo, Federico and Cid-Ruiz, Yairon and Li, Binglin and Monta\ no, Jonathan and Zhang, Naizhen , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2020.107382 , URL =

2020
[244]

Hyry, Eero , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0002-9947-99-02143-1 , URL =

1999 doi
[245]

Brodmann, M. P. and Sharp, R. Y. , TITLE =. 1998 , PAGES =. doi:10.1017/CBO9780511629204 , URL =

1998 doi
[246]

Tohoku Math

Sturmfels, Bernd , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 1991 , NUMBER =. doi:10.2748/tmj/1178227496 , URL =

1991
[247]

Vanishing theorems for combinatorial geometries , author =
[248]

Conca, Aldo and De Negri, Emanuela and Gorla, Elisa , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2020 , NUMBER =. doi:10.1093/imrn/rny075 , URL =

2020 doi
[249]

Conca, Aldo and De Negri, Emanuela and Gorla, Elisa , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2015 , NUMBER =. doi:10.1093/imrn/rnu032 , URL =

2015 doi
[250]

Vietnam J

Conca, Aldo and De Negri, Emanuela and Gorla, Elisa , TITLE =. Vietnam J. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s10013-022-00551-w , URL =

2022 doi
[251]

2024 , note =

Huang, Daoji and Larson, Matt , title =. 2024 , note =

2024
[252]

2025 , note =

Cheng, Ronnie , title =. 2025 , note =

2025
[253]

Xie, Qihong , TITLE =. Math. Z. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00209-009-0585-9 , URL =

2010 doi
[254]

Scott and Billera, Louis J

Provan, J. Scott and Billera, Louis J. , TITLE =. Math. Oper. Res. , FJOURNAL =. 1980 , NUMBER =. doi:10.1287/moor.5.4.576 , URL =

1980 doi
[255]

Mehta, V. B. and Ramanathan, A. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971368 , URL =

1985 doi
[256]

Algebra Number Theory , FJOURNAL =

Payne, Sam , TITLE =. Algebra Number Theory , FJOURNAL =. 2009 , NUMBER =. doi:10.2140/ant.2009.3.107 , URL =

2009 doi
[257]

Katz, Eric , TITLE =. Expo. Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.exmath.2008.04.003 , URL =

2009 doi
[258]

Liu, Ruizhen , title =
[259]

, TITLE =

Brion, M. , TITLE =. Transform. Groups , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/BF01234659 , URL =

1997 doi
[260]

Goresky, Mark and Kottwitz, Robert and MacPherson, Robert , TITLE =. Invent. Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/s002220050197 , URL =

1998 doi
[261]

Topology , FJOURNAL =

Atiyah, Michael and Bott, Raoul , TITLE =. Topology , FJOURNAL =. 1984 , NUMBER =. doi:10.1016/0040-9383(84)90021-1 , URL =

1984 doi
[262]

Segal, Graeme , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1968 , PAGES =

1968
[263]

2015 , PAGES =

Conrad, Brian and Gabber, Ofer and Prasad, Gopal , TITLE =. 2015 , PAGES =. doi:10.1017/CBO9781316092439 , URL =

2015 doi
[264]

Larson, Matt , TITLE =. J. Algebra , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.jalgebra.2025.02.033 , URL =

2025 doi
[265]

, HOWPUBLISHED =

Kov\'acs, S\'andor J. , HOWPUBLISHED =. Cohen-. https://mathoverflow.net/q/247270 , URL =
[266]

Silvotti, Roberto , TITLE =. Invent. Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/s002220050096 , URL =

1996 doi
[267]

1992 , PAGES =

Orlik, Peter and Terao, Hiroaki , TITLE =. 1992 , PAGES =. doi:10.1007/978-3-662-02772-1 , URL =

1992 doi
[268]

Tevelev, Jenia , TITLE =. Amer. J. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1353/ajm.2007.0029 , URL =

2007
[269]

Acta Math

Baum, Paul and Fulton, William and MacPherson, Robert , TITLE =. Acta Math. , FJOURNAL =. 1979 , NUMBER =. doi:10.1007/BF02392091 , URL =

1979 doi
[270]

Topology , FJOURNAL =

Atiyah, Michael and Bott, Raoul and Shapiro, Arnold , TITLE =. Topology , FJOURNAL =. 1964 , NUMBER =. doi:10.1016/0040-9383(64)90003-5 , URL =

1964 doi
[271]

Anderson, Dave and Payne, Sam , TITLE =. Doc. Math. , FJOURNAL =. 2015 , PAGES =

2015
[272]

Thomason, R. W. , TITLE =. Algebraic topology and algebraic. 1987 , ISBN =

1987
[273]

, TITLE =

Weibel, Charles A. , TITLE =. 2013 , PAGES =. doi:10.1090/gsm/145 , URL =

2013 doi
[274]

Kato, Kazuya and Saito, Takeshi , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2004 , PAGES =. doi:10.1007/s10240-004-0026-6 , URL =

2004 doi
[275]

Lucas, Dean , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1975 , PAGES =. doi:10.2307/1997156 , URL =

1975 doi
[276]

Andreas , TITLE =

Nielsen, H. Andreas , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1974 , PAGES =

1974
[277]

1973 , PAGES =

Some theorems on actions of algebraic groups , JOURNAL =. 1973 , PAGES =. doi:10.2307/1970915 , URL =

1973 doi
[278]

1976 , NUMBER =

Some properties of the decompositions of algebraic varieties determined by actions of a torus , JOURNAL =. 1976 , NUMBER =

1976
[279]

2016 , PAGES =

Eisenbud, David and Harris, Joe , TITLE =. 2016 , PAGES =. doi:10.1017/CBO9781139062046 , URL =

2016 doi
[280]

1999 , PAGES =

Flenner, Hubert and O'Carroll, Liam and Vogel, Wolfgang , TITLE =. 1999 , PAGES =. doi:10.1007/978-3-662-03817-8 , URL =

1999 doi
[281]

Toh. On the. Trans. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/tran/8351 , URL =

2021 doi
[282]

On ideals generated by

Toh. On ideals generated by. J. Commut. Algebra , FJOURNAL =. 2021 , NUMBER =. doi:10.1216/jca.2021.13.549 , URL =

2021 doi
[283]

A blowup algebra for hyperplane arrangements , JOURNAL =

Garrousian, Mehdi and Simis, Aron and Toh. A blowup algebra for hyperplane arrangements , JOURNAL =. 2018 , NUMBER =. doi:10.2140/ant.2018.12.1401 , URL =

2018 doi
[284]

Burity, Ricardo , TITLE =. Bull. Sci. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.bulsci.2023.103359 , URL =

2023
[285]

, TITLE =

Stanley, Richard P. , TITLE =. Higher combinatorics (. 1977 , ISBN =

1977
[286]

The geometry of zonotopal algebras

Crowley, Colin and Proudfoot, Nicholas , date =. The geometry of zonotopal algebras
[287]

The omega invariant of a matroid , AUTHOR =
[288]

and Roth, B

Asimow, L. and Roth, B. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1978 , PAGES =. doi:10.2307/1998867 , URL =

1978 doi
[289]

Matroid theory (

Whiteley, Walter , TITLE =. Matroid theory (. 1996 , ISBN =. doi:10.1090/conm/197/02540 , URL =

1996 doi
[290]

, TITLE =

Martin, Jeremy L. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2003 , NUMBER =. doi:10.1090/S0002-9947-03-03321-X , URL =

2003 doi
[291]

Yu, Josephine , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.jcta.2016.11.007 , URL =

2017 doi
[292]

Jackson, Bill and Owen, J. C. , TITLE =. Discrete Appl. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.dam.2017.12.009 , URL =

2019 doi
[293]

Counting fibres of the

Clarke, Oliver and Dewar, Sean and Gallet, Matteo and Grasegger, Georg and Tripp, Daniel Green and Smith, Ben , note=. Counting fibres of the
[294]

A tropical approach to rigidity: counting realisations of frameworks , author=
[295]

On generic rigidity in the plane , JOURNAL =

Lov\'asz, L\'. On generic rigidity in the plane , JOURNAL =. 1982 , NUMBER =. doi:10.1137/0603009 , URL =

1982 doi
[296]

Capco, Jose and Gallet, Matteo and Grasegger, Georg and Koutschan, Christoph and Lubbes, Niels and Schicho, Josef , TITLE =. SIAM J. Appl. Algebra Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1137/17M1118312 , URL =

2018 doi
[297]

and Vidunas, Raimundas , TITLE =

Bartzos, Evangelos and Emiris, Ioannis Z. and Vidunas, Raimundas , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00454-022-00370-3 , URL =

2022 doi
[298]

Bocci, Cristiano and Carlini, Enrico and Kileel, Joe , TITLE =. J. Algebra , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jalgebra.2015.10.008 , URL =

2016 doi
[299]

Topology and geometry---

Mn\"ev, Nikolai , TITLE =. Topology and geometry---. 1988 , ISBN =. doi:10.1007/BFb0082792 , URL =

1988 doi
[300]

2003 , PAGES =

Lafforgue, Laurent , TITLE =. 2003 , PAGES =. doi:10.1090/crmm/019 , URL =

2003 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Aguiar, Marcelo and Ardila, Federico , TITLE =. Mem. Amer. Math. Soc. (to appear) , YEAR =

[2] [2]

Discrete Comput

Ardila, Federico and Benedetti, Carolina and Doker, Jeffrey , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00454-009-9232-9 , URL =

doi:10.1007/s00454-009-9232-9 2010

[3] [3]

Ardila, Federico and Castillo, Federico and Eur, Christopher and Postnikov, Alexander , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =

2020

[4] [4]

Ardila, Federico and Denham, Graham and Huh, June , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2023 , NUMBER =. doi:10.1090/jams/1009 , URL =

doi:10.1090/jams/1009 2023

[5] [5]

Valuations for matroid polytope subdivisions , JOURNAL =

Ardila, Federico and Fink, Alex and Rinc\'. Valuations for matroid polytope subdivisions , JOURNAL =. 2010 , NUMBER =. doi:10.4153/CJM-2010-064-9 , URL =

doi:10.4153/cjm-2010-064-9 2010

[6] [6]

Adiprasito, Karim and Huh, June and Katz, Eric , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2018 , NUMBER =. doi:10.4007/annals.2018.188.2.1 , URL =

doi:10.4007/annals.2018.188.2.1 2018

[7] [7]

Asgari, Mahdi , TITLE =. Canad. J. Math. , FJOURNAL =. 2002 , NUMBER =. doi:10.4153/CJM-2002-025-8 , URL =

work page doi:10.4153/cjm-2002-025-8 2002

[8] [8]

Bastidas, Jose , TITLE =. Algebr. Comb. , FJOURNAL =. 2021 , NUMBER =

2021