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Log-concave poset inequalities.(English)Zbl 1547.05035

Summary: We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave inequalities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions.
In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non-commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley’s inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

MSC:

05A20 Combinatorial inequalities
06A07 Combinatorics of partially ordered sets
06A06 Partial orders, general
05B35 Combinatorial aspects of matroids and geometric lattices

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[1]K. Adiprasito, J. Huh and E. Katz, Hodge theory for combinatorial geometries, Annals of Math. 188 (2018), 381-452. 55, 73, 138, 143 ·Zbl 1442.14194
[2]K. Adiprasito and R. Sanyal, Whitney numbers of arrangements via measure concentration of intrinsic volumes, preprint (2016), 9 pp.; arXiv:1606.09412. 138, 142
[3]A. C. Aitken, The monomial expansion of determinantal symmetric functions, Proc. Roy. Soc. Edin-burgh, Sect. A 61 (1943), 300-310. 140 ·Zbl 0063.00032
[4]A. D. Alexandrov, Zur Theorie der gemischten Volumina von konvexen Körpern IV (in German), Mat. Sbornik 3 (1938), 227-251. 74, 141, 142 ·JFM 64.1347.01
[5]N. Alon, The number of spanning trees in regular graphs, Random Structures Algorithms 1 (1990), 175-181. 145 ·Zbl 0820.05033
[6]N. Alon and J. H. Spencer, The probabilistic method (fourth ed.), John Wiley, Hoboken, NJ, 2016, 375 pp. ·Zbl 1333.05001
[7]N. Anari, K. Liu, S. Oveis Gharan and C. Vinzant, Log-concave polynomials III: Mason’s Ultra-log-concavity conjecture for independent sets of matroids, preprint (2018), 11 pp.; arXiv:1811.01600. 56, 73, 138, 139, 143
[8]N. Anari, K. Liu, S. Oveis Gharan and C. Vinzant, Log-concave polynomials II: High-dimensional walks and an FPRAS for counting bases of a matroid, Annals of Math., to appear; extended abstract in Proc. 51-st STOC, ACM, New York, 2019, 1-12. 73, 138 ·Zbl 1433.68606
[9]F. Ardila, G. Denham and J. Huh, Lagrangian geometry of matroids, Jour. AMS 36 (2023), 727-794. 73, 138 ·Zbl 1512.05068
[10]S. Backman, C. Eur and C. Simpson, Simplicial generation of Chow rings of matroids, Jour. EMS, to appear, 37 pp.; arXiv:1905.07114. 73 ·Zbl 1447.05046
[11]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[12]M. O. Ball and J. S. Provan, Calculating bounds on reachability and connectedness in stochastic networks, Networks 13 (1983), 253-278. 140 ·Zbl 0569.68053
[13]E. F. Beckenbach and R. Bellman, Inequalities (Second ed.), Springer, New York, 1965, 198 pp. 140 ·Zbl 0128.27401
[14]C. Benedetti, R. S. González D’León, C. R. H. Hanusa, P. E. Harris, A. Khare, A. H. Morales and M. Yip, A combinatorial model for computing volumes of flow polytopes, Trans. AMS 372 (2019), 3369-3404. 141 ·Zbl 1420.05011
[15]C. Berge, Hypergraphs, North-Holland, Amsterdam, 1989, 255 pp. 139 ·Zbl 0674.05001
[16]F. Bergeron, G. Labelle and P. Leroux, Combinatorial species and tree-like structures, Cambridge Univ. Press, Cambridge, UK, 1998, 457 pp. 140 ·Zbl 0888.05001
[17]A. Berget, H. Spink and D. Tseng, Log-concavity of matroid h-vectors and mixed Eulerian numbers, Duke Math. J., to appear; preprrint (2020), 29 pp.; arXiv:2005.01937. 73, 138, 140
[18]O. Bernardi, T. Kálmán and A. Postnikov, Universal Tutte polynomial, Adv. Math. 402 (2022), Paper No. 108355, 74 pp. 139 ·Zbl 1487.05130
[19]N. Biggs, Algebraic graph theory, Cambridge Univ. Press, London, 1974, 170 pp. 144 ·Zbl 0284.05101
[20]A. Björner, The unimodality conjecture for convex polytopes, Bull. AMS 4 (1981), 187-188. 142 ·Zbl 0458.52004
[21]A. Björner and M. Wachs, Permutation statistics and linear extensions of posets, J. Combin. Theory A 58 (1991), 85-114. 141 ·Zbl 0742.05084
[22]A. Björner and G. Ziegler, Introduction to greedoids, in Matroid applications, Cambridge Univ. Press, Cambridge, UK, 1992, 284-357. 67, 140 ·Zbl 0772.05026
[23]T. Braden, J. Huh, J. P. Matherne, N. Proudfoot and B. Wang, A semi-small decomposition of the Chow ring of a matroid, Adv. Math. 409 (2022), Paper No. 108646, 49 pp. 73, 143 ·Zbl 1509.14012
[24]T. Braden, J. Huh, J. P. Matherne, N. Proudfoot and B. Wang, Singular Hodge theory for combinatorial geometries, preprint (2020), 95 pp.; arXiv:2010.06088. 73, 143
[25]P. Brändén, Unimodality, log-concavity, real-rootedness and beyond, in Handbook of enumerative combinatorics, CRC Press, Boca Raton, FL, 2015, 437-483. 54, 137, 141 ·Zbl 1327.05051
[26]P. Brändén and J. Huh, Hodge-Riemann relations for Potts model partition functions, preprint (2018), 7 pp.; arXiv:1811.01696. 73, 138
[27]P. Brändén and J. Huh, Lorentzian polynomials, Annals of Math. 192 (2020), 821-891. 56, 62, 73, 74, 85, 138, 139, 140, 143
[28]P. Brändén and J. Leake, Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture, preprint (2021), 10 pp.; arXiv:2110.00487. 143
[29]P. Brändén, J. Leake and I. Pak, Lower bounds for contingency tables via Lorentzian polynomials, Israel Journal of Math., to appear, 28 pp.; arXiv:2008.05907. 73
[30]F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. AMS 81 (1989), no. 413, 106 pp. 54, 137, 141, 146 ·Zbl 0697.05011
[31]F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, in Jerusalem combinatorics, AMS, Providence, RI, 1994, 71-89. 54, 137, 139
[32]G. Brightwell and D. West, Partially ordered sets, Ch. 11 in Handbook of discrete and combinatorial mathematics, CRC Press, Boca Raton, FL, 2000, 717-752. 141
[33]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[34]G. Brightwell and P. Winkler, Counting linear extensions, Order 8 (1991), 225-247. 145 ·Zbl 0759.06001
[35]T. Brylawski, The broken-circuit complex, Trans. AMS 234 (1977), 417-433. 138 ·Zbl 0368.05022
[36]T. Brylawski, The Tutte polynomial. I. General theory, in Matroid theory and its applications, Liguori, Naples, 1982, 125-275. 138, 139
[37]T. Brylawski and J. Oxley, The Tutte polynomial and its applications, in Matroid applications, Cambridge Univ. Press, Cambridge, UK, 1992, 123-225. 138 ·Zbl 0769.05026
[38]Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer, Berlin, 1988, 331 pp. 141 ·Zbl 0633.53002
[39]S. H. Chan and I. Pak, Introduction to the combinatorial atlas, Expo. Math. 40 (2022), 1014-1048. 73, 143 ·Zbl 1504.05027
[40]S. H. Chan and I. Pak, Correlation inequalities for linear extensions, preprint (2022), 23 pp.; arXiv:2211.16637. 69
[41]S. H. Chan and I. Pak, Linear extensions of finite posets, preprint (2023), 55 pages. 125
[42]S. H. Chan, I. Pak and G. Panova, Extensions of the Kahn-Saks inequality for posets of width two, Combinatorial Theory 3 (2023), no. 1, Paper No. 8, 34 pp. 74, 125, 141, 142
[43]S. H. Chan, I. Pak and G. Panova, The cross-product conjecture for width two posets, Trans. AMS 375 (2022), 5923-5961. 74 ·Zbl 1496.05010
[44]S. H. Chan, I. Pak and G. Panova, Effective poset inequalities, SIAM J. Discrete Math. 37 (2023), 1842-1880. ·Zbl 1520.05017
[45]Y. Choe, J. Oxley, A. Sokal and D. Wagner, Homogeneous multivariate polynomials with the half-plane property, Adv. Appl. Math. 32 (2004), 88-187. 85 ·Zbl 1054.05024
[46]S. Chmutov, Topological Tutte Polynomial, in Handbook on the Tutte Polynomial and Related Topics, to appear, CRC Press, Boca Raton, FL, 2021, 22 pp.; arXiv:1708.08132. 139
[47]F. R. K. Chung, P. C. Fishburn and R. L. Graham, On unimodality for linear extensions of partial orders, SIAM J. Algebraic Discrete Methods 1 (1980), 405-410. 141 ·Zbl 0501.06005
[48]D. Cordero-Erausquin, B. Klartag, Q. Merigot and F. Santambrogio, One more proof of the Alexandrov-Fenchel inequality, C.R. Math. Acad. Sci. Paris 357 (2019), no. 8, 676-680. 141 ·Zbl 1428.52013
[49]J. E. Dawson, A collection of sets related to the Tutte polynomial of a matroid, in Lecture Notes in Math. 1073, Springer, Berlin, 1984, 193-204 138, 139 ·Zbl 0544.05014
[50]J. A. De Loera, Y. Kemper and S. Klee, h-vectors of small matroid complexes, Electron. J. Combin. 19 (2012), no. 1, Paper 14, 11 pp. 139 ·Zbl 1244.05238
[51]C. Defant and N. Kravitz, Friends and strangers walking on graphs, Comb. Theory 1 (2021), Paper No. 6, 34 pp. ·Zbl 1498.05148
[52]P. Dembowski, Finite geometries, Springer, Berlin, 1968, 375 pp. 139 ·Zbl 0159.50001
[53]S. Dittmer and I. Pak, Counting linear extensions of restricted posets, preprint (2018), 33 pp.; arXiv:1802.06312. 141, 145
[54]J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, 69-87. 139 ·Zbl 0268.05019
[55]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[56]C. Eur and J. Huh, Logarithmic concavity for morphisms of matroids, Adv. Math. 367 (2020), 107094, 19 pp. 60, 139 ·Zbl 1437.05039
[57]T. Feder and M. Mihail, Balanced Matroids, in 24th STOC (1992), ACM, New York, 26-38. 138
[58]W. Feit, The degree formula for the skew-representations of the symmetric group, Proc. AMS 4 (1953), 740-744. 140 ·Zbl 0052.02302
[59]S. Felsner and L. Wernisch, Markov chains for linear extensions, the two-dimensional case, in Proc. 8th SODA (1997), 239-247. 141 ·Zbl 1321.68369
[60]P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge Univ. Press, Cambridge, 2009, 810 pp. 140, 141 ·Zbl 1165.05001
[61]W. Fulton, Young tableaux, Cambridge Univ. Press, Cambridge, UK, 1997, 260 pp. 140 ·Zbl 0878.14034
[62]R. S. González D’León, C. R. H. Hanusa, A. H. Morales and M. Yip, Column convex matrices, G-cyclic orders, and flow polytopes, Discrete Comput. Geom. 70 (2023), 1593-1631. 141 ·Zbl 1529.52010
[63]G. Gordon and E. McMahon, Interval partitions and activities for the greedoid Tutte polynomial, Adv. Appl. Math. 18 (1997), 33-49. 144 ·Zbl 0906.05014
[64]I. Gorodezky and I. Pak, Generalized loop-erased random walks and approximate reachability, Random Structures Algorithms 44 (2014), 201-223. 139, 140 ·Zbl 1303.05181
[65]J. Gregor, On quadratic Hurwitz forms, Apl. Mat. 26 (1981), 142-153. 85 ·Zbl 0457.15016
[66]G. Grimmett, The random-cluster model, Springer, Berlin, 2006, 377 pp. 139 ·Zbl 1122.60087
[67]H. Guo and M. Jerrum, A polynomial-time approximation algorithm for all-terminal network reliability, SIAM J. Comput. 48 (2019), 964-978. 140 ·Zbl 1430.68441
[68]L. Gurvits, A short proof, based on mixed volumes, of Liggett’s theorem on the convolution of ultra-logconcave sequences, Electron. J. Combin. 16 (2009), no. 1, Note 5, 5 pp. 142 ·Zbl 1159.05054
[69]R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), 697-712. 142 ·Zbl 0658.10001
[70]G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities (second ed.), Cambridge Univ. Press, Cambridge, UK, 1952, 324 pp. 140 ·Zbl 0047.05302
[71]O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972), 190-243. 138 ·Zbl 0228.05131
[72]J. Herzog and T. Hibi, Discrete polymatroids, J. Algebraic Combin. 16 (2002), 239-268. 139 ·Zbl 1012.05046
[73]S. G. Hoggar, Chromatic polynomials and logarithmic concavity, J. Combin. Theory, Ser. B 16 (1974), 248-254. 140 ·Zbl 0268.05104
[74]H. Huang, Induced subgraphs of hypercubes and a proof of the sensitivity conjecture, Annals of Math. 190 (2019), 949-955. 143 ·Zbl 1427.05116
[75]J. Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, Jour. AMS 25 (2012), 907-927. 73, 138, 140 ·Zbl 1243.14005
[76]J. Huh, h-vectors of matroids and logarithmic concavity, Adv. Math. 270 (2015), 49-59. 73, 138, 140 ·Zbl 1304.05013
[77]J. Huh, Combinatorial applications of the Hodge-Riemann relations, in Proc. ICM Rio de Janeiro, vol. IV, World Sci., Hackensack, NJ, 2018, 3093-3111. 54, 137, 138
[78]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[79]J. Huh and E. Katz, Log-concavity of characteristic polynomials and the Bergman fan of matroids, Math. Ann. 354 (2012), 1103-1116. 73, 138 ·Zbl 1258.05021
[80]J. Huh, B. Schröter and B. Wang, Correlation bounds for fields and matroids, J. Eur. Math. Soc. 24 (2022), 1335-1351. 55, 73 ·Zbl 1485.05023
[81]J. Huh and B. Wang, Enumeration of points, lines, planes, etc., Acta Math. 218 (2017), 297-317. 73 ·Zbl 1386.05021
[82]C. Ikenmeyer and I. Pak, What is in #P and what is not?, preprint (2022), 82 pp.; extended abstract in Proc. 63rd FOCS (2022); arXiv:2204.13149. 145
[83]M. Jerrum, Two remarks concerning balanced matroids, Combinatorica 26 (2006), 733-742. 139 ·Zbl 1121.05027
[84]J. Kahn, Some non-Sperner paving matroids, Bull. LMS 12 (1980), 268. 139 ·Zbl 0415.05017
[85]J. Kahn and M. Saks, Balancing poset extensions, Order 1 (1984), 113-126. ·Zbl 0561.06004
[86]A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Order 8 (1991), 7-15. ·Zbl 0736.06002
[87]Yu. Kempner and V. E. Levit, Geometry of poset antimatroids, Electron. Notes Discrete Math. 40 (2013), 169-173. 140
[88]B. Korte, L. Lovász and R. Schrader, Greedoids, Springer, Berlin, 1991, 211 pp. 67, 140, 144 ·Zbl 0733.05023
[89]C. Krattenthaler, Combinatorial proof of the log-concavity of the sequence of matching numbers, J. Combin. Theory, Ser. A 74 (1996), 351-354. 145 ·Zbl 0847.05077
[90]J. P. S. Kung, The geometric approach to matroid theory, in Gian-Carlo Rota on combinatorics, Birkhäuser, Boston, MA, 1995, 604-622; available at tinyurl.com/yds5pcbm 142
[91]J. P. S. Kung, Letter to I. Pak (October 18, 2021), available at tinyurl.com/apu73nzw 146
[92]M. Lenz, Matroids and log-concavity, preprint (2011), 9 pp.; arXiv:1106.2944. 55, 139
[93]M. Lenz, The f -vector of a representable-matroid complex is log-concave, Adv. Appl. Math. 51 (2013), 543-545. 138 ·Zbl 1301.05382
[94]A. Lichnerowic, Géométrie des groupes de transformations (in French), Dunod, Paris, 1958, 193 pp. 142 ·Zbl 0096.16001
[95]T. M. Liggett, Ultra logconcave sequences and negative dependence, J. Combin. Theory, Ser. A 79 (1997), 315-325. 142 ·Zbl 0888.60013
[96]A. W. Marcus, D. A. Spielman and N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Annals of Math. 182 (2015), 307-325. 143 ·Zbl 1316.05066
[97]J. H. Mason, Matroids: unimodal conjectures and Motzkin’s theorem, in Proc. Conf. Combin. Math., Inst. Math. Appl., Southend-on-Sea, UK, 1972, 207-220; available at tinyurl.com/7w7wjz6v 55, 138
[98]P. Matthews, Generating a random linear extension of a partial order, Ann. Probab. 19 (1991), 1367-1392. ·Zbl 0728.60009
[99]I. Mező, Combinatorics and number theory of counting sequences, CRC Press, Boca Raton, FL, 2020, 479 pp. 65 ·Zbl 1445.05001
[100]S. Murai, T. Nagaoka and A. Yazawa, Strictness of the log-concavity of generating polynomials of matroids, J. Combin. Theory, Ser. A 181 (2021), Paper 105351, 22 pp. 57, 61, 73, 140 ·Zbl 1464.05035
[101]K. Murota, Discrete convex analysis, SIAM, Philadelphia, PA, 2003, 389 pp. 139 ·Zbl 1029.90055
[102]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[103]O’H90] K. M. O’Hara, Unimodality of Gaussian coefficients: a constructive proof, J. Combin. Theory, Ser. A 53 (1990), 29-52. 137 ·Zbl 0697.05002
[104]J. Oxley, Matroid theory, Oxford Univ. Press, Oxford, 1992, 532 pp. 138, 139, 142 ·Zbl 0784.05002
[105]I. Pak, Lectures on discrete and polyhedral geometry, monograph draft (2009), 440 pp.; available at math.ucla.edu/˜pak/book.htm
[106]I. Pak, Combinatorial inequalities, Notices AMS 66 (2019), 1109-1112; an expanded version of the paper is available at tinyurl.com/py8sv5v6 74, 141, 145 ·Zbl 1423.05028
[107]I. Pak, What is a combinatorial interpretation?, preprint (2022), 58 pp.; to appear in Open Problems in Algebraic Combinatorics, AMS, Providence, RI; arXiv:2209.06142. 145
[108]I. Pak and G. Panova, Strict unimodality of q-binomial coefficients, C.R. Math. Acad. Sci. Paris 351 (2013), 415-418. 137 ·Zbl 1272.05217
[109]A. Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not., no. 6 (2009), 1026-1106. 139 ·Zbl 1162.52007
[110]R. A. Proctor, Solution of two difficult combinatorial problems with linear algebra, Amer. Math. Monthly 89 (1982), 721-734. 137, 138 ·Zbl 0509.05007
[111]R. C. Read, An introduction to chromatic polynomials, J. Combin. Theory 385 4 (1968), 52-71. 140 ·Zbl 0173.26203
[112]R. C. Read and G. F. Royle, Chromatic roots of families of graphs, in Graph theory, combinatorics, and applications, vol. 2, Wiley, New York, 1991, 1009-1029. 142 ·Zbl 0841.05034
[113]R. C. Read and W. T. Tutte, Chromatic polynomials, in Selected topics in graph theory, vol. 3, Academic Press, San Diego, CA, 1988, 15-42. 142 ·Zbl 0667.05022
[114]G.-C. Rota, Combinatorial theory, old and new, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, 1971, 229-233. 138 ·Zbl 0362.05044
[115]G.-C. Rota and L. H. Harper, Matching theory, an introduction, in Advances in Probability and Related Topics, vol. 1, Dekker, New York, 1971, 169-215. 138 ·Zbl 0234.05001
[116]A. Saumard and J. A. Wellner, Log-concavity and strong log-concavity: a review, Stat. Surv. 8 (2014), 45-114. 137 ·Zbl 1360.62055
[117]R. Schneider, Convex bodies: the Brunn-Minkowski theory (second ed.), Cambridge Univ. Press, Cambridge, UK, 2014, 736 pp. 141 ·Zbl 1287.52001
[118]A. Schrijver, Combinatorial optimization. Polyhedra and efficiency, vols. A-C, Springer, Berlin, 2003, 1881 pp. 61 ·Zbl 1041.90001
[119]Y. Shenfeld and R. van Handel, Mixed volumes and the Bochner method, Proc. AMS 147 (2019), 5385-5402. 74, 85, 87, 141, 142, 143 ·Zbl 1457.52008
[120]Y. Shenfeld and R. van Handel, The extremals of the Alexandrov-Fenchel inequality for convex polytopes, Acta Math. 231 (2023), 89-204. 70, 74, 142, 143, 144 ·Zbl 1529.05032
[121]OEIS] N. J. A. Sloane, The online encyclopedia of integer sequences, oeis.org 65, 70
[122]A. D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in Surveys in combinatorics, Cambridge Univ. Press, Cambridge, UK, 2005, 173-226. 139 ·Zbl 1110.05020
[123]R. P. Stanley, Unimodal sequences arising from Lie algebras, in Lecture Notes in Pure and Applied Math. 57, Dekker, New York, 1980, 127-136. 138 ·Zbl 0451.05004
[124]JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 ·Zbl 1547.05035
[125]R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), 168-184. 138 ·Zbl 0502.05004
[126]R. P. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory, Ser. A 31 (1981), 56-65. 69, 73, 74, 141 ·Zbl 0484.05012
[127]R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph theory and its applications, New York Acad. Sci., New York, 1989, 500-535. 54, 137
[128]R. P. Stanley, Enumerative Combinatorics, vol. 1 (second ed.) and vol. 2, Cambridge Univ. Press, 2012 and 1999. 140, 141 ·Zbl 0928.05001
[129]R. P. Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, Paper 9, 24 pp. ·Zbl 1169.06002
[130]D. Stanton, Unimodality and Young’s lattice, J. Combin. Theory, Ser. A 54 (1990), 41-53. 142 ·Zbl 0736.05009
[131]J. R. Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge, Trans. AMS 359 (2007), 1115-1128. 142 ·Zbl 1110.06009
[132]J. J. Sylvester, Proof of the hitherto undemonstrated Fundamental Theorem of Invariants, Philosophical Magazine 5 (1878), 178-188; available at tinyurl.com/c94pphj 137 ·JFM 10.0082.02
[133]B. Teissier, Bonnesen-type inequalities in algebraic geometry. I. Introduction to the problem, in Seminar on Differential Geometry, Princeton Univ. Press, Princeton, NJ, 1982, 85-105. 141 ·Zbl 0494.52009
[134]W. T. Trotter, Partially ordered sets, in Handbook of combinatorics, vol. 1, Elsevier, Amsterdam, 1995, 433-480. 141 ·Zbl 0841.06001
[135]J. H. van Lint, The van der Waerden conjecture: two proofs in one year, Math. Intelligencer 4 (1982), no. 2, 72-77. 141 ·Zbl 0503.15009
[136]D. J. A. Welsh, Matroid theory, Academic Press, London, 1976, 433 pp. 59, 139, 142 ·Zbl 0343.05002
[137]D. J. A. Welsh, Complexity: knots, colourings and counting, Cambridge Univ. Press, Cambridge, UK, 1993, 163 pp. 145 ·Zbl 0799.68008
[138]H. Weyl, Über die Starrheit der Eiflächen und konvexer Polyeder (in German), Berl. Ber. (1917), 250-266. 142 ·JFM 46.1115.02
[139]D. B. Wilson, Generating random spanning trees more quickly than the cover time, in Proc. 28-th STOC, ACM, New York, 1996, 296-303. 140 ·Zbl 0946.60070
[140]P. M. Winkler, Correlation and order, in Combinatorics and ordered sets, AMS, Providence, RI, 1986, 151-174. 141 ·Zbl 0598.06001
[141]AUTHORS Swee Hong Chan Department of Mathematics, Rutgers University, New Brunswick, NJ, USA sweehong [dot] chan [at] rutgers [dot] edu https://sites.math.rutgers.edu/ sc2518 JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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