Asymptotic behavior of the number of geodesics in the discrete Heisenberg group
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 39-52 Cet article a éte moissonné depuis la source Math-Net.Ru

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A. M. Vershik; A. V. Malyutin. Asymptotic behavior of the number of geodesics in the discrete Heisenberg group. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 39-52. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a3/

[1] A. M. Vershik, “Dinamicheskaya teoriya rosta v gruppakh: entropiya, granitsy, primery”, UMN, 55:4(334) (2000), 59–128 | DOI | MR | Zbl

[2] A. M. Vershik, “Zadacha o tsentralnykh merakh na prostranstvakh putei graduirovannykh grafov”, Funkts. anal. i ego pril., 48:4 (2014), 26–46 | DOI | MR | Zbl

[3] A. M. Vershik, “Osnaschennye graduirovannye grafy, proektivnye predely simpleksov i ikh granitsy”, Zap. nauchn. semin. POMI, 432, 2015, 83–104 | Zbl

[4] A. M. Vershik, A. V. Malyutin, “Fazovyi perekhod v zadache o granitse-vykhod dlya sluchainykh bluzhdanii na gruppakh”, Funkts. anal. i ego pril., 49:2 (2015), 7–20 | DOI | MR | Zbl

[5] A. M. Vershik, A. V. Malyutin, “Beskonechnye geodezicheskie v diskretnoi gruppe Geizenberga”, Zap. nauchn. semin. POMI, 462, 2017, 39–51

[6] A. M. Vershik, A. V. Malyutin, “Absolyut konechno porozhdennykh grupp: II. Laplasova i vyrozhdennaya chasti”, Funkts. anal. i ego pril., 52:3 (2018), 3–21 | DOI | MR | Zbl

[7] E. Yu. Smirnov, “Diagrammy Yunga i $q$-kombinatorika”, Kvant, 2015, no. 1, 7–12

[8] R. Stenli, Perechislitelnaya kombinatorika, Mir, M., 1990

[9] Dzh. Endryus, Teoriya razbienii, Nauka, M., 1982

[10] G. Almkvist, G. E. Andrews, “A Hardy–Ramanujan formula for restricted partitions”, J. Number Theory, 38:2 (1991), 135–144 | DOI | MR | Zbl

[11] G. E. Andrews, “On the difference of successive Gaussian polynomials”, J. Statist. Plann. Inference, 34:1 (1993), 19–22 | DOI | MR | Zbl

[12] M. Braun, “An algebraic interpretation of the $q$-binomial coefficients”, Int. Electron. J. Algebra, 6 (2009), 23–30 | MR | Zbl

[13] F. Brenti, “Log-concave and unimodal sequences in algebra, combinatorics, and geometry: An update”, Jerusalem Combinatorics'93, Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994, 71–89 | DOI | MR | Zbl

[14] L. M. Butler, “A unimodality result in the enumeration of subgroups of a finite abelian group”, Proc. Amer. Math. Soc., 101:4 (1987), 771–775 | DOI | MR | Zbl

[15] L. M. Butler, “The $q$-log-concavity of $q$-binomial coefficients”, J. Combin. Theory Ser. A, 54:1 (1990), 54–63 | DOI | MR | Zbl

[16] H. Cohn, “Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients”, Amer. Math. Monthly, 111:6 (2004), 487–495 | MR | Zbl

[17] S. DeSalvo, I. Pak, “Log-concavity of the partition function”, Ramanujan J., 38:1 (2015), 61–73 | DOI | MR | Zbl

[18] V. Dhand, Rank-unimodality of Young's lattice via explicit chain decomposition, 2013, arXiv: 1303.2536

[19] V. Dhand, “Tropical decomposition of Young's partition lattice”, J. Algebraic Combin., 39:4 (2014), 783–806 | DOI | MR | Zbl

[20] V. Dhand, “A combinatorial proof of strict unimodality for $q$-binomial coefficients”, Discrete Math., 335 (2014), 20–24 | DOI | MR | Zbl

[21] F. Goodman, K. M. O'Hara, “On the Gaussian polynomials”, $q$-Series and Partitions, IMA Vol. Math. Appl., 18, Springer, New York, 1989, 57–66 | MR

[22] V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York, 2002 | MR | Zbl

[23] D. E. Knuth, “Subspaces, subsets, and partitions”, J. Combin. Theory Ser. A, 10 (1971), 178–180 | DOI | MR | Zbl

[24] T. Konstantopoulos, L. Yuan, “A probabilistic interpretation of the Gaussian binomial coefficients”, J. Appl. Probab., 54:4 (2017), 1295–1298 | DOI | MR | Zbl

[25] J. Konvalina, “Generalized binomial coefficients and the subset-subspace problem”, Adv. Appl. Math., 21:2 (1998), 228–240 | DOI | MR | Zbl

[26] J. Konvalina, “A unified interpretation of the binomial coefficients, the Stirling numbers, and the Gaussian coefficients”, Amer. Math. Monthly, 107:10 (2000), 901–910 | DOI | MR | Zbl

[27] C. Krattenthaler, “On the $q$-log-concavity of Gaussian binomial coefficients”, Monatsh. Math., 107:4 (1989), 333–339 | DOI | MR | Zbl

[28] I. G. Macdonald, “An elementary proof of a $q$-binomial identity”, $q$-Series and Partitions, IMA Vol. Math. Appl., 18, Springer, New York, 1989, 73–75 | MR

[29] K. M. O'Hara, “Unimodality of Gaussian coefficients: a constructive proof”, J. Combin. Theory Ser. A, 53:1 (1990), 29–52 | DOI | MR | Zbl

[30] I. Pak, G. Panova, “Strict unimodality of $q$-binomial coefficients”, C. R. Math. Acad. Sci. Paris, 351 (2013), 415–418 | DOI | MR | Zbl

[31] I. Pak, G. Panova, “Bounds on certain classes of Kronecker and $q$-binomial coefficients”, J. Combin. Theory Ser. A, 147 (2017), 1–17 | DOI | MR | Zbl

[32] R. A. Proctor, “Solution of two difficult combinatorial problems with linear algebra”, Amer. Math. Monthly, 89:10 (1982), 721–734 | DOI | MR | Zbl

[33] V. Reiner, D. Stanton, “Unimodality of differences of specialized Schur functions”, J. Algebraic Combin., 7:1 (1998), 91–107 | DOI | MR | Zbl

[34] B. E. Sagan, “Inductive and injective proofs of log concavity results”, Discrete Math., 68:2–3 (1988), 281–292 | DOI | MR | Zbl

[35] B. E. Sagan, “Inductive proofs of $q$-log concavity”, Discrete Math., 99:1–3 (1992), 289–306 | DOI | MR | Zbl

[36] B. E. Sagan, “Log concave sequences of symmetric functions and analogs of the Jacobi–Trudi determinants”, Trans. Amer. Math. Soc., 329:2 (1992), 795–811 | DOI | MR | Zbl

[37] M. Shapiro, “A geometric approach to the almost convexity and growth of some nilpotent groups”, Math. Ann., 285 (1989), 601–624 | DOI | MR | Zbl

[38] R. P. Stanley, “Weyl groups, the hard Lefschetz theorem, and the Sperner property”, SIAM J. Algebraic Discrete Methods, 1:2 (1980), 168–184 | DOI | MR | Zbl

[39] R. P. Stanley, “Some aspects of groups acting on finite posets”, J. Combin. Theory Ser. A, 32:2 (1982), 132–161 | DOI | MR | Zbl

[40] R. P. Stanley, “Log-concave and unimodal sequences in algebra, combinatorics, and geometry”, Graph Theory and its Applications: East and West, Ann. New York Acad. Sci., 576, New York Acad. Sci., New York, 1989, 500–535 | DOI | MR | Zbl

[41] R. P. Stanley, F. Zanello, “Some asymptotic results on $q$-binomial coefficients”, Ann. Comb., 20:3 (2016), 623–634 | DOI | MR | Zbl

[42] J. J. Sylvester, “Proof of the hitherto undemonstrated Fundamental Theorem of Invariants”, Philosophical Magazine, 5 (1878), 178–188 ; reprinted in Coll. Math. Papers, v. 3, Chelsea, New York, 1973, 117–126 | Zbl

[43] L. Takács, “Some asymptotic formulas for lattice paths”, J. Statist. Plann. Inference, 14:1 (1986), 123–142 | DOI | MR | Zbl

[44] A. M. Vershik, “Intrinsic metric on graded graphs, standardness, and invariant measures”, Zap. nauchn. semin. POMI, 421, 2014, 58–67 | MR | Zbl

[45] A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi) groups”, European J. Math., 4:4 (2018), 1476–1490 | DOI | MR | Zbl

[46] D. E. White, “Monotonicity and unimodality of the pattern inventory”, Adv. Math., 38:1 (1980), 101–108 | DOI | MR | Zbl

[47] D. Zeilberger, “A one-line high school algebra proof of the unimodality of the Gaussian polynomials $n\brack k$ for $k20$”, $q$-Series and Partitions, IMA Vol. Math. Appl., 18, Springer, New York, 1989, 67–72 | MR

[48] D. Zeilberger, “Kathy O'Hara's constructive proof of the unimodality of the Gaussian polynomials”, Amer. Math. Monthly, 96:7 (1989), 590–602 | DOI | MR | Zbl

[49] F. Zanello, “Zeilberger's KOH theorem and the strict unimodality of $q$-binomial coefficients”, Proc. Amer. Math. Soc., 143:7 (2015), 2795–2799 | DOI | MR | Zbl

[50] F. Zanello, “On Bergeron's positivity problem for $q$-binomial coefficients”, Electron. J. Combin., 25:2 (2018), Paper 2.17 | MR | Zbl